The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality

Gerhard Huisken^{1} and Tom Ilmanen^{2}

In this paper we develop the theory of weak solutions for the inverse mean curvature flow of hypersurfaces in a Riemannian manifold, and apply it to prove the Riemannian Penrose inequality for a connected horizon, to wit: the total mass of an asymptotically flat 3-manifold of nonnegative scalar curvature is bounded below in terms of the area of each smooth, compact, connected, “outermost” minimal surface in the 3-manifold. A minimal surface is called outermost if it is not separated from infinity by any other compact minimal surface. The result was announced in [HI1].

Let M be a smooth Riemannian manifold of dimension n ≥ 2 with metric g = (g_{ij}). A classical solution of the inverse mean curvature flow is a smooth family x : N ×[0, T] → M of hypersurfaces N_{t} := x(N, t) satisfying the parabolic evolution equation

(⋆)  \frac{∂x}{∂t} = \frac{ν}{H} ,  x ∈ N_{t},  0 ≤ t ≤ T,

where H, assumed to be positive, is the mean curvature of N_{t} at the point x, ν is the outward unit normal, and ∂x/∂t denotes the normal velocity field along the surface N_{t}.

Without special geometric assumptions (see [Ge,U,Ho]), the mean curvature may tend to zero at some points and singularities develop. In §1 we introduce a level-set formulation of (⋆), where the evolving surfaces are given as level-sets of a scalar function u via

N_{t} = ∂{x : u(x) < t},

and (⋆) is replaced by the degenerate elliptic equation

(⋆⋆)  div_{M} (\frac{∇u}{|∇u|}) = |∇u|,

This document was prepared using AMS-TEX. 1Mathematisches Institut, Auf der Morgenstelle 10, D-72076 Tübingen, Germany. Partially supported by Sonderforschungsbereich 382, Tübingen. 2Department of Mathematics, Northwestern University, Evanston IL 60201, USA, and Max-Planck Institut für Mathematik, Inselstrasse 22-26, D-04103 Leipzig, Germany. Partially supported by an NSF Postdoctoral Fellowship, NSF Grants DMS-9626405 and DMS-9708261, and a Sloan Foundation Fellowship. where the left hand side describes the mean curvature of the level-sets and the right hand side yields the inverse speed. This formulation in divergence form admits locally Lipschitz continuous solutions and is inspired by the work of Evans-Spruck [ES] and Chen-Giga-Goto [CGG] on the mean curvature flow.

Using a minimization principle similar to those of Lichnewsky-Temam [LT], Luckhaus [Lu], Zhou and Hardt-Zhou [Z,HZ], and Visintin [V], together with elliptic regularization, in §2-3 we prove existence and uniqueness of a Lipschitz continuous weak solution of (⋆⋆) having level-sets of bounded nonnegative weak mean curvature. The existence result (Theorem 3.1) assumes only mild conditions on the underlying manifold M, with no restrictions on dimension. The minimization principle used in the definition of weak solutions of (⋆⋆) allows the evolving surfaces to jump instantaneously over a positive 3-volume (momentary “fattening”), which is desirable for our main application. See Pasch [Pa] for numerical computation of this phenomenon.

The minimization property is also essential for the uniqueness, compactness and regularity properties of the solution. Furthermore, it implies connectedness of the evolving surfaces (§4), which is essential in our argument.

The Riemannian Penrose Inequality. In §5-§8 we employ the inverse mean curvature flow in asymptotically flat 3-manifolds to prove the Riemannian Penrose Inequality as stated below.

An end of a Riemannian 3-manifold (M, g) is called asymptotically flat if it is realized by an open set that is diffeomorphic to the complement of a compact set K in R^{3}, and the metric tensor g of M satisfies

(0.1)  |g_{ij} − δ_{ij}| ≤ \frac{C}{|x|},  |g_{ij,k}| ≤ \frac{C}{|x|^{2}} ,

as |x| → ∞. The derivatives are taken with respect to the Euclidean metric δ = (δ_{ij}) on R^{3} \ K. In addition, we require the Ricci curvature of M to satisfy

(0.2)  Rc ≥ −\frac{Cg}{|x|^{2}} .

Following Arnowitt, Deser and Misner [ADM] the ADM mass, or total energy, of the end is defined by a flux integral through the sphere at infinity,

(0.3)  m := lim_{r→∞} \frac{1}{16π} ∫_{∂B_{r}^{δ}(0)}(g_{ii,j} − g_{ij,i})n^{j} dμ_{δ}.

This is a geometric invariant of the given end, despite being expressed in coordinates. It is finite precisely when the the scalar curvature R of g satisfies

∫_{M} |R| < ∞.

See Bartnik [B] and Chruściel [Ch3] for these facts, detailed below in Lemma 7.3, and see [AM,CK,F,Wa1,Y] for further discussion of asymptotic flatness.

The Riemannian case of the Positive Mass Theorem, first proven by Schoen and Yau [SY3], states that an aymptotically flat 3-manifold of nonnegative scalar curvature (possibly with a compact, minimal boundary) has

m ≥ 0,

with equality only in the case of Euclidean space. A number of other proofs and approaches to this theorem have appeared in the physics and mathematics literature: these include Schoen-Yau [SY2,SY9-SY11] by the method of minimal surfaces and conformal changes; Geroch [G] and Jang [J1-J3] introducing the inverse mean curvature flow; Jang [J4], Kijowski [K1,K2], Jezierski-Kijowski [JK1,JK2], Chruściel [Ch1,Ch2,Ch4,Ch6], and Jezierski [Je1-Je3] using the p-Laplacian; Witten [W], Parker-Taubes [PT], Reula [R], Choquet-Bruhat [C], Reula-Tod [RT], and Chruściel [Ch5] using harmonic spinors; Penrose-Sorkin-Woolgar [PSW] using a single complete null geodesic; Lohkamp [Lo] for a recent Riemannian geometry approach; and others. See also the survey article by Lee-Parker [LP]. Such analytic methods have many additional consequences for the geometry and topology of positive scalar curvature manifolds; see [SY1,SY5,SY6] and [Lc,GL1-GL3].

In 1973, motivated by a physical argument (see below), Penrose [P2] conjectured that

16πm^{2} ≥ |N|,

where |N| is the area of a minimal surface in M. Partial or related results have been obtained by Gibbons [G1,G4] in the case of thin shells using quermassintegrals, Herzlich [He] via the Dirac operator, Bartnik [Ba6] for metrics with quasi-spherical foliations, Jezierski [Je1,Je3] in certain cases using the 5-Laplacian, and Bray [B1] using isoperimetric regions. For further information about the role of isoperimetric regions in positive mass, see [CY,HY].

The following theorem is the main result of this paper.

Main Theorem (Riemannian Penrose Inequality). Let M be a complete, connected 3-manifold. Suppose that

(i) M has nonnegative scalar curvature,

(ii) M is asymptotically flat satisfying (0.1) and (0.2), with ADM mass m, (iii) The boundary of M is compact and consists of minimal surfaces, and M contains

no other compact minimal surfaces.

Then

(0.4)  m ≥ \sqrt{\frac{|N|}{16π}} ,

where |N| is the area of any connected component of ∂M. Equality holds if and only if M is isometric to one-half of the spatial Schwarzschild manifold.

The spatial Schwarzschild manifold is the manifold R^{3} \ {0} equipped with the metric g := (1 + m/2|x|)^{4}δ. Note that it possesses an inversive isometry fixing the sphere ∂B_{m/2}(0), which is an area-minimizing sphere of area 16πm^{2}. The manifold M to which the theorem applies is then R^{3} \ B_{m/2}(0). It is the standard spacelike slice of the exterior region of the Schwarzschild solution of the Einstein vacuum equations, describing a static, rotationally symmetric black hole.

The second clause of condition (iii) seems restrictive at first. Suppose, however, that M is asymptotically flat and has compact, minimal boundary. Then (iii) is satisfied by the infinite portion of M that stands outside the union of all compact minimal surfaces in M (Lemma 4.1). Accordingly, a connected, asymptotically flat 3-manifold that satisfies conditions (i)-(iii) is called an exterior region. An exterior region has the topology of R^{3} minus a finite number of balls, and its boundary consists of area-minimizing 2-spheres. These facts, well known in the literature, are collected in section §4.

Condition (iii) ensures that ∂M is not shielded from spacelike infinity by a smaller minimal surface further out. It is easy to construct a rotationally symmetric example of positive scalar curvature in which the inequality holds for the “outermost” minimal surface, but fails for arbitrarily large minimal surfaces hidden beyond the outermost one. (See figure.) For an example of this shielding effect with several boundary components, see Gibbons [G3].

Large hidden surfaces.

Our result allows ∂M to be disconnected, but only applies to each connected component of ∂M. This limitation seems to be fundamental, due to the use of the Gauss-Bonnet formula in deriving the Geroch Monotonicity formula.^{3}

Physical Interpretation. The manifold (M, g) arises as a spacelike hypersurface in an asymptotically flat Lorentzian manifold L^{3,1} modelling a relativistic isolated gravitating system governed by Einstein’s equations. In this setting, the result can be interpreted as an optimal lower bound for the total energy of the system measured at spacelike infinity in terms of the size of the largest black hole contained inside. The positive scalar curvature of M corresponds to the physical hypothesis of nonnegative energy density on L, expressed by the so-called weak energy condition ([HE,Wa1]), provided that M is a maximal hypersurface in L (see [Ba1]). If M is totally geodesic in L (at least along N), then the minimal surface N corresponds to a so-called closed trapped surface that presages the formation of a space-

3Using a novel and surprising technique, Bray [B2] has recently proven the most general form of the Riemannian Penrose Inequality, namely 16πm^{2} ≥ |∂M| with no connectedness requirement on ∂M. This is a considerable strengthening. He employs a quasistatic flow that couples a moving surface with an evolving conformal factor; it bears a family resemblance to the Hele-Shaw flow. time singularity and ultimately a black hole [HP,HE,Wa1]. The shielding effect corresponds to causal separation.

Penrose originally conjectured (0.4) as a consequence of a physical argument. He assumed that M could be taken as the momentarily static (i.e. totally geodesic) initial slice of a spacetime L with a well-defined, asymptotically flat lightlike infinity existing for all time, and invoked various physical hypotheses and principles including the strong or the dominant energy condition, the Hawking-Penrose Singularity Theorems, the Weak Cosmic Censorship Hypothesis, the Hawking Area Monotonicity Theorem, long-term settling to a stationary black hole, and the No-Hair Theorem (classification of stationary black holes), which he collectively referred to as the “establishment viewpoint” of black hole formation. From these he deduced a chain of inequalities leading to the purely Riemannian assertion (0.4) for the 3-manifold M. For details of this beautiful argument, see [P2,P3,JW], as well as relevant discussion in [H1,G,H3,G2,G3,To,Wa1].

Of these ingredients, the most controversial is the famous (Weak) Cosmic Censorship Hypothesis [P1,P3,Wa2], which asserts that in a generic asymptotically flat spacetime, any singularity is surrounded by a well-defined event horizon, so that it cannot be seen from lightlike infinity. In particular, the singularity cannot have any causal effects on the outside world. Penrose viewed (0.4) as a Riemannian test of Cosmic Censorship. Thus our result rules out a class of potential counterexamples to this important conjecture.

For further information on the physical interpretation, see the announcement [HI1]. See also the informative review article of Penrose [P3].

Outline of Proof. We now sketch our proof of the Penrose Inequality. (A more detailed overview appears in the announcement [HI1].) Hawking [H2] introduced the Hawking quasi-local mass of a 2-surface, defined by

m_{H}(N) := \frac{|N|^{1/2}}{(16π)^{3/2}} (16π − ∫_{N} H^{2}) ,

and observed that it approaches the ADM mass for large coordinate spheres. Geroch [G] introduced the inverse mean curvature flow and discovered that the Hawking mass of the evolving surface is monotone nondecreasing, provided that the surface is connected and the scalar curvature of M is nonnegative. This gave an argument in favor of positive mass. Jang-Wald [JW] realized that if there were a classical solution of the flow starting at the inner boundary and approaching large coordinate spheres as t → ∞, the Geroch Monotonicity result would imply the Penrose Inequality.

The heart of our proof (§5) is to justify the Geroch Monotonicity Formula (5.8) for our weak solutions, even in the presence of jumps. In this step, it is crucial that the evolving surface start and remain connected, as was established in §4. In §6, we extend the Monotonicity Formula to each connected component of a multi-component boundary by exploiting the ability of the evolving surface to jump over regions in 3-space.

Section 7 establishes that the Hawking mass of the evolving surface is eventually bounded by the ADM mass, leading to the proof of the Main Theorem in §8.

The positive integrand on the right-hand side of the Geroch Monotonicity Formula can serve as a kind of “mass density” on M. It is noncanonical in that it depends on the evolving family of surfaces. Remarkably, it vanishes precisely on standard expanding spheres in flat space and Schwarzschild – thus the rigidity statement of the Main Theorem. We revisit the Schwarzschild example in §8 to show that the imputed mass distribution – in particular, whether it resides in the black hole or is distributed throughout the field surrounding it – depends on where you start the flow.

Though noncanonical, the “mass density” can often be estimated in terms of the local geometry of M, leading to explicit lower bounds for the ADM mass. We exploit this idea in §9 to study the quasi-local mass (or gravitational capacity) proposed by Bartnik [Ba3,Ba4,Ba5]. We prove two conjectures of Bartnik: first, the Bartnik capacity is positive on any nonflat fragment of space, and second, the ADM mass is the supremum of the Bartnik capacities of all compact subsets. We end with conjectures that refine those of Bartnik [Ba3,Ba5] about the convergence and existence of 3-manifolds that minimize the ADM mass.

In a future paper we will prove smoothness of the flow outside some finite region, convergence of the Hawking quasi-local mass to the ADM mass, and asymptotic convergence to the center of mass in the sense of Huisken-Yau [HY], under suitable asymptotic regularity for (M, g).

The Full Penrose Conjecture. Inequality (0.4) is subsumed under the space-time form of the Penrose Conjecture, in which the assumption on the extrinisic curvature of M within L is dropped. This allows M to carry nonzero momentum and angular momentum, and would generalize the space-time version of the positive mass theorem (see Schoen-Yau [SY4,SY7,SY8]). Let E_{ij} := R_{ij}^{L} − R^{L}g_{ij}^{L}/2 be the Einstein tensor of L. We say that L satisfies the dominant energy condition if E(m, n) ≥ 0 for all future timelike vectors m, n.

Conjecture. Let N be an “outermost” spacelike 2-surface with null mean curvature vector in an asymptotically flat Lorentzian manifold L satisfying the dominant energy condition. Then

(0.5)  16πm^{2} ≥ |N|,

where m is the mass of L.

The mass of L is defined as \sqrt{E^{2} − P_{1}^{2} − P_{2}^{2} − P_{3}^{2}}, where (E, P^{1}, P^{2}, P^{3}) is the ADM energy-momentum 4-vector of L (see [Wa1, p.293]). For the definition of asymptotically flat for a spacetime L, see [AM,CK,F,Wa1,Y].

A major puzzle is the correct formulation of “outermost”. The careful 4-dimensional definition and analysis of a trapping horizon in Hayward [Ha] may provide clues. An obvious possibility is to require the existence of an asymptotically flat, spacelike slice extending from N to infinity and containing no compact surface (besides the boundary) with nonspacelike mean curvature vector. Of course, such a slice may be useful in proving the inequality. See Jang [J2], Schoen-Yau [SY8] for interesting and relevant calculations.

In this connection, it is worth noting that the Hawking mass is also monotone under the codimension-two inverse mean curvature flow

\frac{∂x}{∂t} = −\frac{H⃗}{|H|^{2}} , x ∈ N_{t}, t ≥ 0.

of a spacelike 2-surface with spacelike mean curvature vector in a Lorentz manifold satisfying the dominant energy condition, as can be verified by calculation. Unfortunately, this 2x2 system of evolution equations is backward-forward parabolic.

There are sharper versions of the Penrose Inequality for charged black holes [J3] and rotating black holes [P3, p.663]. The former can now be rigourously established using our Geroch Monotonicity Formula (5.24), as was pointed out to us by G. Gibbons, while the latter would require understanding space-time issues like those in the above conjecture.

Another important case treated by Penrose’s original argument involves an infinitely thin shell of matter collapsing at the speed of light, surrounding a region of flat Minkowski space. Gibbons [G1,G4] reduces this to the Minkowski-like inequality

∫_{N} |H⃗| ≥ 4\sqrt{π}|N|^{1/2}

for a spacelike 2-surface in R^{3,1} with inner-pointing, spacelike mean curvature vector H⃗, which he then establishes using recent work of Trudinger [Tr] on quermassintegrals. Accordingly, the Penrose Inequality has often been called the Isoperimetric Inequality for black holes.

Acknowledgements. We would like to thank B. Andrews, R. Bartnik, G. Belletini, P. Chruściel, M. Friedman, R. Geroch, G. Gibbons, R. Hamilton, M. Herzlich, J. Isenberg, P. Li, R. Schoen, R. M. Wald, S.-T. Yau, and R. Ye for stimulating and useful conversations. The paper started at the CIME meeting in Cetraro and the PDG meeting in Oberwolfach, in summer 1996.

§1 Variational Formulation of Inverse Mean Curvature Flow

Because the inverse mean curvature flow does not remain smooth, we give a weak formulation based on minimizing a functional. In general, the surfaces satisfying the weak formulation are C^{1,α}, but jump discontinuously across a set of positive measure when a certain minimizing criterion ceases to be met. Nevertheless, the area of the surface evolves continuously, except possibly at t = 0.

Notation. Let N be a smooth hypersurface in the Riemannian manifold M with metric g. We write h = (h_{ij}) for the induced metric on N, ∇ for the connection on M, D for the induced connection on N, and Δ for the Laplace-Beltrami operator on N.

Suppose N = ∂E where E is an open set in M, and let ν = (ν^{i}) be the outward unit normal to E along N. Define the second fundamental form A = (A_{ij}^{N}) of N by

A(e_{1}, e_{2}) := ⟨∇_{e_{1}}ν, e_{2}⟩, e_{1}, e_{2} ∈ T_{x}N, x ∈ N,

mean curvature scalar H := h^{ij}A_{ij}, and mean curvature vector H⃗ := −Hν. For a standard ball, H > 0 and H⃗ points inward.

Elementary Observations. A simple example of inverse mean curvature flow is the expanding sphere ∂B_{R(t)} in R^{n}, where

R(t) := e^{t/(n−1)}.

Gerhardt [Ge] proved that any compact, starshaped initial surface remains star-shaped and smooth under the flow, and becomes an expanding round sphere as t → ∞. See Gerhardt [Ge] or Urbas [U] for similar results for more general expanding flows. Some noncompact self-similar examples are given in [HI2].

Note that (⋆) is invariant under the transformation

x ↦ λx, t ↦ t,

so t is unitless. This suggests that we will not get much regularity in time, and that any singularities encountered will have a purely spatial character.

Let (N_{t})_{0≤t<T} be a smooth family of hypersurfaces satisfying (⋆). Write v for the outward normal speed. By the first variation formula (see (1.11)), the area element evolves in the normal direction by

(1.1)  \frac{∂}{∂t}dμ_{t} = Hv dμ_{t} = dμ_{t}.

Here (∂/∂t)dμ_{t} is the Lie derivative of the form dμ_{t} along a bundle of trajectories orthogonal to N_{t}. Therefore the area satisfies

\frac{d}{dt}|N_{t}| = |N_{t}|,  |N_{t}| = e^{t}|N_{0}|,  t ≥ 0.

Next we derive the evolution of the mean curvature. For a general surface moving with normal speed v, by taking the trace of the Riccati equation, we get

(1.2)  \frac{∂H}{∂t} = Δ(−v) − |A|^{2}v − Rc(ν, ν)v,

which yields for (⋆),

(1.3)  \begin{aligned}\frac{∂H}{∂t} & = −Δ (\frac{1}{H} ) − \frac{|A|^{2}}{H} − \frac{Rc(ν, ν)}{H} \\ & = \frac{ΔH}{H^{2}} − \frac{2|DH|^{2}}{H^{3}} − \frac{|A|^{2}}{H} − \frac{Rc(ν, ν)}{H}, \end{aligned}

where Rc is the Ricci curvature of M. This equation is cause for optimism, because in view of the fact that |A|^{2} ≥ H^{2}/(n − 1), the parabolic maximum principle yields the curvature bound

(1.4)  max_{N_{t}} H^{2} ≤ max_{N_{0}} H^{2} + C,

as long as the Ricci curvature is bounded below and the flow remains smooth.

This estimate constrasts sharply with ordinary mean curvature flow (see [Hu]), where the |A|^{2} term has a positive sign, unleashing a menagerie of singularities [I2]. Together with the variational formulation to be given below, the negative |A|^{2} is the key to the regularity theory. Unfortunately, the same term has a tendency to cause H → 0.

By a computation similar to [Hu], we find the following evolution equation for the norm of the second fundamental form

\frac{d}{dt}|A|^{2} = \frac{1}{H^{2}} (Δ|A|^{2} − 2|DA|^{2} + 4DH · \frac{A}{H} · DH + 2|A|^{4} − 4H tr A^{3}

+ A ∗ A ∗ Rm + A ∗ ∇Rm),

where ∗ represents a linear combination of contractions. The |A|^{4}/H^{2} term suggests that the second fundamental form blows up, especially where H ↘ 0.

Torus Counterexample. We give an example to establish that smoothness cannot be preserved. Suppose that N_{0} is a thin torus in R^{3}, obtained as the boundary of an εneighborhood of a large round circle. Then H > 0 initially, and the flow exists for a short time. Now (1.4) yields a uniform lower bound for the speed, so the torus steadily fattens up, and if this were to continue, H would become negative in the donut hole. This shows that N_{t} cannot remain smooth, and suggests that the surface must inexorably change topology, as indeed does occur in our formulation.

Level-Set Description. Because it is desirable that H remain positive (or at least nonnegative), we impose a unidirectional ansatz that compels this. This is accomplished via a level-set formulation, inspired by Evans-Spruck [ES], Chen-Giga-Goto [CGG]. Assume that the flow is given by the level-sets of a function u : M → R via

E_{t} := {x : u(x) < t}, N_{t} := ∂E_{t}.

Wherever u is smooth with ∇u ≠ 0, equation (⋆) is equivalent to

(⋆⋆)  div_{M} (\frac{∇u}{|∇u|}) = |∇u|,

where the left side gives the mean curvature of {u = t} and the right side gives the inverse speed.

Weak Formulation. It appears that (⋆) is not a gradient flow, nor is (⋆⋆) an Euler-Lagrange equation. Yet in the context of geometric measure theory, the mean curvature bound (1.4) calls out for a minimization principle. Freezing the |∇u| term on the right-hand side, consider equation (⋆⋆) as the Euler-Lagrange equation of the functional

J_{u}(v) = J_{u}^{K}(v) := ∫_{K} |∇v| + v|∇u| dx.

The idea of freezing the time derivative and minimizing an elliptic functional is seen in [LT,Lu,HZ,V], where it is the the key to existence, compactness, and regularity.

Definition. Let u be a locally Lipschitz function on the open set Ω. Then u is a weak solution (subsolution, supersolution respectively) of (⋆⋆) on Ω provided

(1.5)  J_{u}^{K}(u) ≤ J_{u}^{K}(v)

for every locally Lipschitz function v (v ≤ u, v ≥ u respectively) such that {v ≠ u} ⊂⊂ Ω, where the integration is performed over any compact set K containing {u ≠ v}. (It does not matter which such set we use, so we will usually drop the K.)

By virtue of the identity

J_{u}(min(v, w)) + J_{u}(max(v, w)) = J_{u}(v) + J_{u}(w),

whenever {v ≠ w} is precompact, applied with w = u, we see that u is a weak solution if and only if u is simultaneously a weak supersolution and a weak subsolution.

Equivalent Formulation. Let us particularize the above definition to individual level sets. If F is a set of locally finite perimeter, let ∂^{∗}F denote its reduced boundary. If a set X is supposed to be (n − 1)-dimensional, write |X| for the (n − 1)-dimensional Hausdorff measure of X. Write EΔF := (E \ F) ∪ (F \ E) for the symmetric difference of sets E and F.

Given K ⊆ M and u locally Lipschitz, define the functional

J_{u}(F) = J_{u}^{K}(F) := |∂^{∗}F ∩ K| − ∫_{F ∩K} |∇u|,

for a set F of locally finite perimeter. We say that E minimizes J_{u} in a set A (on the outside, inside respectively) if

J_{u}^{K}(E) ≤ J_{u}^{K}(F),

for each F such that F ΔE ⊂⊂ A (with F ⊇ E, F ⊆ E respectively), and any compact set K containing F ΔE. Again, the choice of such K does not matter, and we will generally drop K in the sequel.

In general, we have

H^{n−1}⌊∂^{∗}(E ∪ F) + H^{n−1}⌊∂^{∗}(E ∩ F) ≤ H^{n−1}⌊∂^{∗}E + H^{n−1}⌊∂^{∗}F,

as Radon measures, so

(1.6)  J_{u}(E ∪ F) + J_{u}(E ∩ F) ≤ J_{u}(E) + J_{u}(F),

whenever EΔF is precompact. (The inequality reflects possible cancellation of oppositely oriented pieces.) Therefore E minimizes J_{u} (in A) if and only if E minimizes J_{u} on the inside and on the outside (in A).

1.1 Lemma. Let u be a locally Lipschitz function in the open set Ω. Then u is a weak solution (subsolution, supersolution respectively) of (⋆) in Ω if and only if for each t, E_{t} := {u < t} minimizes J_{u} in Ω (on the outside, inside respectively).

The reader may look ahead to Example 1.5 to gain some insight into the effect of this minimization principle.

Proof of Lemma 1.1. 1. Let v be a locally Lipschitz function such that {v ≠ u} ⊂⊂ Ω and K a compact set containing {v ≠ u}. Set E_{t} := {u < t}, F_{t} := {v < t} and note that F_{t}ΔE_{t} ⊆ K for every t. Select a < b such that a < u, v < b on K, and compute by the co-area formula (see [S,p.70]),

(1.7)  \begin{aligned}J_{u}^{K}(v) & = ∫_{K} |∇v| + v|∇u| \\ & = ∫_{a}^{b} |∂^{∗}F_{t} ∩ K| − ∫_{K} ∫_{a}^{b} χ_{\{v(x)<t\}}|∇u| + b ∫_{K} |∇u| \\ & = ∫_{a}^{b} J_{u}^{K}(F_{t}) + b ∫_{K} |∇u|. \end{aligned}

If each E_{t} minimizes J_{u} in Ω, this formula shows that u minimizes (1.5) in Ω, and is thereby a weak solution of (⋆⋆). The same argument treats weak supersolutions and subsolutions separately. This proves one direction of the Lemma.

2. Now suppose that u is a supersolution of (⋆⋆). Fix t_{0} and F such that

F ⊆ E_{t_{0}}, E_{t_{0}} \ F ⊂⊂ Ω.

We aim to show that J_{u}(E_{t_{0}}) ≤ J_{u}(F). Since J_{u} is lower semicontinuous, we may assume that

(1.8)  J_{u}(F) ≤ J_{u}(G),

for all G with GΔE_{t_{0}} ⊆ F ΔE_{t_{0}}. Now define the nested family

F_{t} := \{ \begin{matrix}F ∩ E_{t}, & t ≤ t_{0}, \\ E_{t}, & t > t_{0}. \end{matrix}

By (1.8), J_{u}(F) ≤ J_{u}(E_{t} ∪ F), so by (1.6),

J_{u}(F_{t}) ≤ J_{u}(E_{t}) for all t.

Define v so that F_{t} = {v < t}, that is,

v := \{ \begin{matrix}t_{0} & on E_{t_{0}} \ F \\ u & elsewhere. \end{matrix}

Note that v ∈ BV_{loc} ∩ L_{loc}^{∞} and {v ≠ u} ⊂⊂ Ω, so J_{u}(v) makes sense. Approximating v by smooth functions with |∇v_{i}| ⇀ |∇v| as measures, we find that J_{u}(u) ≤ J_{u}(v). Furthermore, (1.7) is valid for v. So by (1.7), we conclude

∫_{a}^{b} J_{u}(E_{t}) dt ≤ ∫_{a}^{b} J_{u}(F_{t}) dt.

Together with the above, this shows that J_{u}(E_{t}) = J_{u}(F_{t}) for a.e. t. By (1.6), this shows J_{u}(E_{t} ∪ F) ≤ J_{u}(F) for a.e. t ≤ t_{0}. Passing t ↗ t_{0}, we obtain by lower semicontinuity

J_{u}(E_{t_{0}}) ≤ J_{u}(F).

We have proven: if u is a supersolution of (⋆⋆), then for each t_{0}, E_{t_{0}} minimizes J_{u} on the inside.

3. Next assume that u is a subsolution of (⋆⋆). Similarly to the above, we prove: for each t, {u ≤ t} minimizes J_{u} on the outside. Choose t_{i} ↗ t_{0}, and note that {u ≤ t_{i}} converges to E_{t_{0}} locally in L^{1}. Using lower semicontinuity of J_{u} and a standard replacement argument, it follows that E_{t_{0}} minimizes J_{u} on the outside.

Initial Value Problem. We will usually combine the above definition with an initial condition consisting of an open set E_{0} with a boundary that is at least C^{1}. We say that u is a weak solution of (⋆⋆) with initial condition E_{0} if

(††)  u ∈ C_{loc}^{0,1}(M), E_{0} = {u < 0}, and u satisfies (1.5) in M \ E_{0}.

This imposes the Dirichlet boundary condition u = 0 on ∂E_{0}. It is reasonable to expect Lipschitz continuity up to the boundary in view of (1.4), provided that the mean curvature of ∂E_{0} is bounded at least in a weak sense. For particularly bad initial conditions, however, this requirement should be weakened (see [HI2]).

Let E_{t} be a nested family of open sets in M, closed under ascending union. Define u by the characterization E_{t} = {u < t}. We say that (E_{t})_{t>0} is a weak solution of (⋆) with initial condition E_{0} if

(†)  u ∈ C_{loc}^{0,1}(M) and E_{t} minimizes J_{u} in M \ E_{0} for each t > 0.

Observe that (†) allows slightly more competitors than (††), since it permits F ΔE_{t} to touch ∂E_{0}. Nevertheless, using Lemma 1.1 and approximating up to the boundary, it is quite straightforward to show

1.2 Lemma. (††) is equivalent to (†).

At this point, the crucial question arises to whether the equation holds at the initial time. By approximating s ↘ t, we see that (††) and (†) are equivalent to

(1.9)  u ∈ C_{loc}^{0,1}(M) and {u ≤ t} minimizes J_{u} in Ω \ E_{0} for each t ≥ 0.

However, we cannot deduce that E_{0} minimizes J_{u} in M \ E_{0}, since in general {u = 0} may have positive measure. In fact, the equation can be satisfied at t = 0 only if E_{0} is assumed to be in the correct class to begin with! We will take up this essential point in Lemma 1.4 below.

Regularity. Our variational formulation was partly inspired by Visintin [V], who noticed that an appropriate minimization principle can force H to be bounded. As a result, the regularity theory is essentially spatial in character, and succumbs to methods of geometric measure theory. The price of keeping H bounded is temporal irregularity in the form of jumps.

Let f be a bounded measurable function on a domain Ω with smooth metric g. Suppose E contains an open set U and minimizes the functional

|∂^{∗}F | + ∫_{F} f

with respect to competitors F such that

F ⊇ U, F ΔE ⊂⊂ Ω.

We say that E minimizes area plus bulk energy f in Ω, respecting the obstacle U.

The following regularity theorem is obtained by [Mi,E] in the C^{1} case; [Ma,A,Ln,T1] in the C^{1,α} case, 0 < α < 1, and [BK,Ln,SWZ] in the C^{1,1} case. In the references for (ii), the case of a nonflat background metric g is not treated. However, carefully reworking the monograph [T2] suffices to establish the estimate with the given dependencies.

1.3 Regularity Theorem. Let n < 8.

(i) If ∂U is C^{1} then ∂E is a C^{1} submanifold of Ω.

(ii) If ∂U is C^{1,α}, 0 < α ≤ 1/2, then ∂E is a C^{1,α} submanifold of Ω. The C^{1,α} estimates depend only on the distance to ∂Ω, ess sup |f|, C^{1,α} bounds for ∂U, and C^{1} bounds (including positive lower bounds) for the metric g.

(iii) If ∂U is C^{2} and f = 0 (the case of pure area minimization with obstacle), then ∂E is C^{1,1}, and C^{∞} where it does not contact the obstacle U.

Let u solve (††). For the duration of the paper, set

E_{t} := {u < t},  E_{t}^{+} := int{u ≤ t},  N_{t} := ∂E_{t},  N_{t}^{+} := ∂E_{t}^{+}.

For n < 8, the Regularity Theorem, Lemma 1.2, and (1.9) imply that N_{t}^{+} = ∂{u > t} and that N_{t} and N_{t}^{+} possess locally uniform C^{1,α} estimates depending only on the local Lipschitz bounds for u.^{4} In particular, for all t > 0,

(1.10)  N_{s} → N_{t} as s ↗ t,  N_{s} → N_{t}^{+} as s ↘ t,

4M. Heidusch [Hd] has recently improved this to C^{1,1} regularity. in the sense of local C^{1,β} convergence, 0 ≤ β < α. If ∂E_{0} is C^{1,α}, the estimates and convergence also hold as s ↘ 0.

When n ≥ 8, the Regularity Theorem and the convergence (1.10) remain true away from a closed singular set Z of dimension at most n − 8 and disjoint from Ū.

Weak Mean Curvature. In order to give a PDE interpretation of our solutions, we will define the mean curvature of a surface that is not quite smooth. Let N be a hypersurface in M, X a compactly supported vectorfield defined on M, and (Φ^{s})_{−ε<s<ε} the flow of diffeomorphisms generated by X, Φ^{0} = id_{M}. The first variation formula for area [S, p.80] states

(1.11)  \frac{d}{ds}|_{s=0} |Φ^{s}(N) ∩ W | = ∫_{N∩W} div_{N} X dμ = ∫_{N∩W} Hν · X dμ,

for any precompact open set W containing the support of X. Here dμ is the surface measure of N, and div_{N} X(x) := ∑_{i} ∇_{e_{i}}X(x) · e_{i}, where e_{1}, … e_{n−1} is any orthonormal basis of T_{x}N. When N is smooth, the first inequality is proven by differentiating under the integral sign, and the second by integration by parts.

When N is C^{1}, or C^{1} with a small singular set and locally finite Hausdorff measure, we use the second equality as a definition. A locally integrable function H on N is called the mean curvature provided it satisfies (1.11) for every X in C_{c}^{∞}(T M).

For minimizers of (1.5), by the co-area formula and the dominated convergence theorem,

0 = \frac{d}{ds}|_{s=0} J_{u}(u ∘ Φ^{s})

= \frac{d}{ds}|_{s=0} (∫_{−∞}^{∞} ∫_{N_{t}∩W} | det dΦ^{s}(x)| dH^{n−1}(x) dt + ∫_{W} u(Φ^{s}(x))|∇u(x)| dx)

= ∫_{−∞}^{∞} ∫_{N_{t}∩W} div_{N_{t}} X dH^{n−1} dt + ∫_{W} ∇u · X|∇u| dx.

Thus the linear functional of X given by the first term is bounded by C ∫_{M} |X| dx, and therefore can be represented by a vectorfield Hν on M, where H is some bounded measurable function and ν is a measurable unit vectorfield on M. By the co-area formula, we obtain

0 = ∫_{−∞}^{∞} ∫_{N_{t}}(Hν + ∇u) · X dH^{n−1} dt, X ∈ C_{c}^{1}(T M).

By Lebesgue differentiation and comparison with the above, this shows that for a.e. t and a.e. x ∈ N_{t}, ν is the unit normal, H is the weak mean curvature, and

(1.12)  H = |∇u|  a.e. x ∈ N_{t},  a.e. t.

For later use, we mention the following convergence fact. If N^{i} is a sequence of C^{1} hypersurfaces, N^{i} → N locally in C^{1}, and

sup_{i} ess sup_{N^{i}} |H_{N^{i}}| < ∞,

it follows from (1.11) and the Riesz Representation Theorem that H_{N} exists weakly as a locally L^{1} function, with the weak convergence

(1.13)  ∫_{N^{i}} H_{N^{i}}ν_{N^{i}} · X → ∫_{N} H_{N}ν_{N} · X,  X ∈ C_{c}^{0}(T M),

and lower semicontinuity

(1.14)  ess sup_{N} |H_{N}| ≤ lim inf_{i→∞} ess sup_{N_{i}} |H_{N_{i}}|,  ∫_{N} ϕ|H_{N}|^{2} ≤ lim inf_{i→∞} ∫_{N_{i}} ϕ|H_{N_{i}}|^{2},

for any ϕ ∈ C_{c}^{0}(M).

Minimizing Hulls. To explain the jumps, we introduce some terminology. Let Ω be an open set. We call E a minimizing hull (in Ω) if E minimizes area on the outside in Ω, that is, if

|∂^{∗}E ∩ K| ≤ |∂^{∗}F ∩ K|,

for any F containing E such that F \ E ⊂⊂ Ω, and any compact set K containing F \ E. We say that E is a strictly minimizing hull (in Ω) if equality implies that F ∩ Ω = E ∩ Ω a.e. (See [BT] for a related definition.)

It is easily checked that the Lebesgue points of a minimizing hull form an open set in Ω. In general we prefer to work with this unique, open, representatative.

The intersection of a countable collection of minimizing hulls is a minimizing hull, and the same goes for strictly minimizing. Now let E be any measurable set. Define E′ = E′_{Ω} to be the intersection of (the Lebesgue points of) all the strictly minimizing hulls in Ω that contain E. Working modulo sets of measure zero, this may be realized by a countable intersection, so E′ itself is a strictly minimizing hull, and open. We call E′ the strictly minimizing hull of E (in Ω). Note that E′′ = E′.

If E ⊂⊂ M and M grows at infinity (for example, if M is asymptotically flat), then E′, taken in M, is precompact as well. In this case E′ satisfies the “shrink-wrap” obstacle problem

|∂E′| ≤ |∂F | whenever E ⊆ F ⊂⊂ M.

Sometimes there are several distinct solutions, but they are all contained in E′ a.e.

If ∂E is C^{2}, the C^{1,1} regularity given in Theorem 1.3(iii) applies to E′. For the weak mean curvature, we have

(1.15) H_{∂E′} = 0 on ∂E′ \ ∂E,  H_{∂E′} = H_{∂E} ≥ 0 H^{n−1} a.e. on ∂E′ ∩ ∂E.

In this language, our variational definition of the flow has the following important consequence. M need not be complete.

1.4 Minimizing Hull Property. Suppose that u satisfies (††) and M has no compact components. Then

(i) For t > 0, E_{t} is a minimizing hull in M,

(ii) For t ≥ 0, E_{t}^{+} is a strictly minimizing hull in M, (iii) For t ≥ 0, E′_{t} = E_{t}^{+}, provided that E_{t}^{+} is precompact, (iv) For t > 0, |∂E_{t}| = |∂E_{t}^{+}|, provided that E_{t}^{+} is precompact. This extends to t = 0

precisely if E_{0} is a minimizing hull.

Note that the function u can take an arbitrary constant value on each compact component of M that does not meet E_{0}; the resulting nonuniqueness would falsify (ii) and (iii).

Proof. (i) By (†),

(1.16)  |∂E_{t} ∩ K| + ∫_{F \E_{t}} |∇u| ≤ |∂^{∗}F ∩ K|,

for t > 0, any F with F ΔE_{t} ⊂⊂ M \ E_{t}, and K containing F ΔE_{t}. That is, E_{t} is a minimizing hull, t > 0.

(ii) By (1.9) and the fact that |∇u| = 0 a.e. on {u = t},

(1.17)  |∂E_{t}^{+} ∩ K| + ∫_{F \E_{t}^{+}} |∇u| ≤ |∂^{∗}F ∩ K|,

for t ≥ 0, any F with F ΔE_{t}^{+} ⊂⊂ M \E_{t}, and suitable K, proving that E_{t}^{+} is a minimizing hull, t ≥ 0. To prove strictly minimizing, suppose F contains E_{t}^{+} and

|∂^{∗}F ∩ K| = |∂E_{t}^{+} ∩ K|.

Then by (1.17), ∇u = 0 a.e. on F \ E_{t}^{+}. And clearly, F itself is a minimizing hull, so by a measure zero modification, we may assume F is open. Then u is constant on each connected component of F \ Ē_{t}^{+}. Since M has no compact components and F has no wasted perimeter, no such component can have closure disjoint from Ē_{t}^{+}, so u = t on F \ E_{t}^{+}, so F ⊆ E_{t}^{+}. This proves that E_{t}^{+} is a strictly minimizing hull.

(iii) It follows that E′_{t} ⊆ E_{t}^{+}. If E_{t}^{+} is precompact, then E′_{t} must coincide with E_{t}^{+}, for otherwise |∂E′_{t}| < |∂E_{t}^{+}|, contradicting (1.17).

(iv) If E_{t}^{+} is precompact, then so is E_{t}, and by cross substitution in (1.16) and (1.17), we get |∂E_{t}| ≤ |∂E_{t}^{+}| and |∂E_{t}^{+}| ≤ |∂E_{t}|, proving (iv) for t > 0, and for t = 0 if E_{0} happens to be a minimizing hull itself.

Returning to the question of the initial condition, we see by Lemma 1.4(iv) that E_{t} minimizes J_{u} in M \ E_{0} for all t ≥ 0 if and only if (†) holds and E_{0} is a minimizing hull. This gives a stronger attainment of the initial condition than (†) alone. In this case, one might say that (E_{t})_{0≤t<∞} is a weak solution of (⋆). However, this terminology differs so subtly from our previous locutions that we will avoid it.

As a consequence of Lemma 1.4, the effect of the minimization principle (1.5) can be described heuristically as follows. The statement is not precise since, in principle, there may be an infinite number of jump times.

(1) As long as E_{t} remains a minimizing hull, it flows by the usual inverse mean curvature flow.

(2) When this condition is violated, E_{t} jumps to E′_{t} and continues.

In particular, the mean curvature is nonnegative on the flowing surfaces after time zero, even if the curvature of ∂E_{0} is mixed.

Two Spheres Joining.

1.5 Two Spheres Example. In the following example, the surface does indeed jump. Suppose E_{0} is the union of two balls in R^{3} at some distance from one another. Each ball expands until the first instant t = t_{jump} when the two spheres can be enclosed by a connected surface of equal area. At this instant, a catenoidal bridge forms between the two balls and the flow continues.

Note that t_{jump} < t_{touch}, the time when the two spheres would make contact classically. If we flow up to time t_{touch} without jumping, then u minimizes (1.5) on {u < t_{touch}}, but not on larger sets. So, not only does the flow meet the obstacle of its own history, but if it waits too long to jump, it cannot continue at all.

G. Bellettini originally showed us this example, observing that any choice of jump time (before the spheres touch) yields a viscosity solution of (⋆⋆), so viscosity solutions are nonunique. The variational principle J_{u} serves to select the jump time.

A more extreme form of jumping is

(1.18)  if u weakly solves (⋆⋆), then so does min(u, t) for every constant t.

That is, the surfaces would like to jump instantly to infinity in order to exploit the negative bulk energy term in J_{u}(F). To prove this, recall that for s ≤ t and K containing E_{s}ΔF,

|∂E_{s} ∩ K| − ∫_{E_{s}∩K} |∇u| ≤ |∂^{∗}F ∩ K| − ∫_{F ∩K} |∇u|.

If u is replaced by min(u, t), the left hand side only increases, while the right-hand side is unchanged. This shows that min(u, t) is a weak solution, by Lemma 1.1.

1.6 Exponential Growth Lemma. Let (E_{t})_{t>0} solve (†) with initial condition E_{0}. As long as E_{t} remains precompact, we have the following.

(i) e^{−t}|∂E_{t}| is constant for t > 0.

(ii) If E_{0} is a minimizing hull, then |∂E_{t}| = e^{t}|∂E_{0}|.

Proof. It follows from the minimization property (†) that J_{u}(E_{t}) is independent of t, t > 0. By the co-area formula, this implies that |∂E_{t}| − ∫_{0}^{t} |∂E_{s}| ds is constant for t > 0, which yields (i). A similar exponential growth applies to |∂E_{t}^{+}|, t ≥ 0. If E_{0} is a minimizing hull, then Lemma 1.4(iv) yields (ii).

§2 Further Properties of Inverse Mean Curvature Flow

In this section we derive compactness and uniqueness theorems for weak solutions of (1.5), and state the relationship between classical and weak solutions.

2.1 Compactness Theorem. Let u_{i} be a sequence of solutions of (1.5) on open sets Ω_{i} in M such that

u_{i} → u, Ω_{i} → Ω,

locally uniformly, and for each K ⊂⊂ Ω,

sup_{K} |∇u_{i}| ≤ C(K),

for large i. Then u is a solution of (1.5) on Ω.

Remarks. 1. For each t, the Regularity Theorem 1.3(ii) implies that any subsequence of N_{t}^{i} has a further subsequence that converges in C^{1,α} to some hypersurface Ñ, away from the singular set Z for u. Whenever there is no jump, Ñ = N_{t} = N_{t}^{+}, so for the full sequence,

(2.1)  N_{t}^{i} → N_{t}  locally in C^{1,α} in Ω \ Z,

for any t not belonging to the countable set of jump times.

2. The Lemma and Remark 1 still hold if we allow u_{i} to solve (1.5) with respect to a metric g_{i} that converges to g in C_{loc}^{1}.

Proof. 1. Let v be a locally Lipschitz function such that {v ≠ u} ⊂⊂ Ω. We must prove that J_{u}(u) ≤ J_{u}(v). First we assume v < u + 1.

Let ϕ ∈ C_{c}^{1}(Ω) be a cutoff function such that ϕ = 1 on {v ≠ u}. Then

v_{i} := ϕv + (1 − ϕ)u_{i}

is a valid comparison function for u_{i}. So by (1.5), for sufficiently large i,

∫_{U} |∇u_{i}| + u_{i}|∇u_{i}| ≤ ∫_{U} |∇v_{i}| + v_{i}|∇u_{i}|

= ∫_{U} |ϕ∇v + (1 − ϕ)∇u_{i} + ∇ϕ(v − u_{i})| + (ϕv + (1 − ϕ)u_{i})|∇u_{i}|

for appropriate U, so

∫_{U} ϕ|∇u_{i}|(1 + u_{i} − v) ≤ ∫_{U} ϕ|∇v| + |∇ϕ(v − u_{i})|.

The last term converges to zero. Since 1 + u_{i} − v is eventually positive and converges uniformly, it follows by lower semicontinuity that

∫_{U} ϕ|∇u|(1 + u − v) ≤ ∫_{U} ϕ|∇v|,

so u satisfies (1.5) for all v < u + 1.

2. Next assume that u satisfies (1.5) for all w ≤ u + k, and prove that it does for each v ≤ u + 2k. Define

v_{1} := min(v, u + k), v_{2} := max(v − k, u).

Then v_{1}, v_{2} ≤ u + k, so inserting v_{1} into J_{u}(u) ≤ J_{u}(v), we get

∫ |∇u| + u|∇u| ≤ ∫_{v≤u+k} |∇v| + v|∇u| + ∫_{v>u+k} |∇u| + (u + k)|∇u|.

Inserting v_{2} into the same inequality, we get

∫ |∇u| + u|∇u| ≤ ∫_{v≤u+k} |∇u| + u|∇u| + ∫_{v>u+k} |∇v| + (v − k)|∇u|.

Adding these two inequalities and cancelling the extra J_{u}(u), we get J_{v}(u) ≤ J_{u}(v), as required.

2.2 Uniqueness Theorem. Assume that M has no compact component; M need not be complete.

(i) If u and v solve (1.5) on an open set Ω in M, and {v > u} ⊂⊂ Ω, then v ≤ u on Ω.

(ii) If (E_{t})_{t>0} and (F_{t})_{t>0} solve (†) in a manifold M and the initial conditions satisfy E_{0} ⊆ F_{0}, then E_{t} ⊆ F_{t} as long as E_{t} is precompact in M.

(iii) In particular, for a given E_{0}, there exists at most one solution (E_{t})_{t>0} of (†) such that each E_{t} is precompact.

Without the compactness assumption, min(u, t) gives an example of nonuniqueness. Another example occurs in the metric

g := a^{2}(t)dr^{2} + e^{t/2}g_{S^{2}},

where g_{S^{2}} is the standard metric. The evolution is N_{t} = {t − c} × S^{2} quite independent of the choice of a. For a(t) := 1 + 1/t, t > 0, the metric is asymptotic to a cylinder at one end and R^{3} at the other, and a noncompact E_{t} rushes in from infinity at any time t = c. This gives nonuniqueness with E_{0} = ∅.

Proof. (i) First assume that u is a strict weak supersolution of (⋆⋆) in the sense that for any Lipschitz function w ≥ u with {w ≠ u} ⊂⊂ Ω, we have

(2.2)  ∫ |∇u| + u|∇u| + ε ∫ (w − u)|∇u| ≤ ∫ |∇w| + w|∇u|,

where ε > 0. Replace w by u + (v − u)_{+} to obtain

∫_{v>u} |∇u| + u|∇u| + ε ∫_{v>u}(v − u)|∇u| ≤ ∫_{v>u} |∇v| + v|∇u|,

and replace v by v − (v − u)_{+} and u by v in (1.5) to obtain

(2.3)  ∫_{v>u} |∇v| + v|∇v| ≤ ∫_{v>u} |∇u| + u|∇v|.

Adding these, we get

(2.4)  ∫_{v>u}(v − u)(|∇v| − |∇u|) + ε ∫_{v>u}(v − u)|∇u| ≤ 0.

We expect the first integral to be much smaller than the second one near the point of contact, because |∇u| nearly cancels |∇v|. To bring this out, we need something like an upper bound for the positive quantity ∫ (v −u)|∇u|, so we employ the minimizing property of u once again. We replace v by u + (v − s − u)_{+}, s ≥ 0, in (1.5) and integrate over s to obtain

∫_{0}^{∞} ∫_{v−s>u} |∇u| + u|∇u| dx ds ≤ ∫_{0}^{∞} ∫_{v−s>u} |∇v| + (v − s)|∇u| dx ds.

(Since the functional is nonlinear, there can be no action at a distance; the gap must be calibrated.) Switching the order of integration, we have

∫_{M} |∇u| ∫_{−∞}^{∞} χ_{{s>0}}χ_{{v−u>s}}(1 + u − v + s) ds dx ≤ ∫_{M} |∇v| ∫_{−∞}^{∞} χ_{{s>0}}χ_{{v−u>s}} ds dx

which yields

∫_{v>u} −\frac{(v − u)^{2}}{2} |∇u| ≤ ∫_{v>u}(v − u)(|∇v| − |∇u|).

Inserting this into (2.4) yields

∫_{v>u} −\frac{(v − u)^{2}}{2} |∇u| + ε(v − u)|∇u| ≤ 0.

Assuming v ≤ u + ε, the above inequality implies that |∇u| = 0 a.e. on {v > u}, and then (2.3) implies that |∇v| = 0 a.e. on {v > u}. Then u and v are constant on each component of {v > u}. Since {v > u} is precompact in Ω and Ω has no compact component, we conclude that v ≤ u + ε implies v ≤ u.

For general v, by subtracting a constant we can arrange that 0 < sup(v − u) ≤ ε, contradicting what has just been proven. This establishes the result for any u that satisfies (2.2).

Now let u be an arbitrary weak supersolution of (⋆⋆). We may assume that u > 0. Observe that for any ε > 0, u^{ε} := u/(1 − ε) satisfies (2.2) and {v > u^{ε}} is precompact. The above discussion shows that v ≤ u^{ε}, so v ≤ u.

(ii) Recall by (1.18) that v^{t} := min(v, t) solves (1.5) on M \ F̄_{0}. Let W := E_{t} \ F̄_{0}, a precompact open set. Since E_{0} ⊆ F_{0}, we have v^{t} < u + δ near ∂W for each δ > 0, so {v^{t} > u + δ} ⊂⊂ W, so (i) implies v^{t} ≤ u + δ on W, so v^{t} ≤ u on W. Since u < t on W, v ≤ u on W. Thus E_{t} ⊆ F_{t}.

(iii) is immediate from (ii).

The next proposition shows that smooth flows satisfy the weak formulation in the domain they foliate.

2.3 Smooth Flow Lemma. Let (N_{t})_{c≤t<d} be a smooth family of surfaces of positive mean curvature that solves (⋆) classically. Let u = t on N_{t}, u < c in the region bounded by N_{c}, and E_{t} := {u < t}. Then for c ≤ t < d, E_{t} minimizes J_{u} in E_{d} \ E_{c}.

Proof. The exterior normal, defined by ν_{u} := ∇u/|∇u|, is a smooth unit vector field on Ω with div ν_{u} = H_{N_{t}} = |∇u| > 0. Using ν_{u} as a calibration, by the divergence theorem |∂E_{t}| − ∫_{E_{t}} |∇u| = ∫_{∂E} ν_{∂E} · ν_{u} − ∫_{E} |∇u| = ∫_{∂^{∗}F} ν_{∂^{∗}F} · ν_{u} − ∫_{F} |∇u| ≤ |∂^{∗}F | − ∫_{F} |∇u|,

for any finite perimeter set F differing compactly from E_{t}.

The next proposition says that the weak evolution of a smooth, positively curved, strictly minimizing hull is smooth for a short time.

2.4 Smooth Start Lemma. Let E_{0} be a precompact open set in M such that ∂E_{0} is smooth with H > 0 and E_{0} = E′_{0}. Then any weak solution (E_{t})_{0<t<∞} of (†) with initial condition E_{0} coincides with the unique smooth, classical solution for a short time, provided that E_{t} remains precompact for a short time.

Proof. Since (⋆) is parabolic when H > 0, a classical solution F_{t} exists for a short time. By Lemma 2.3, this gives rise to a smooth solution v of (††) defined in an open neighborhood W of E_{0}. By 1.4(iii) and the hypotheses, E_{0}^{+} = E′_{0} = E_{0}. Since E_{t} is precompact, E_{t} converges in Hausdorff distance to E_{0}^{+} as t ↘ 0, which shows that E_{t} is precompact in W for a short time. Then by Theorem 2.2(ii), E_{t} = F_{t} for a short time.

Remark. It can be shown that the weak solution remains smooth until the first moment when either E_{t} ≠ E′_{t}, H ↘ 0, or |A| ↗ ∞. Whether the latter two possibilities occur is unknown.

§3 Elliptic Regularization and Existence

In this section we prove existence of solutions of the initial value problem (††) by means of elliptic regularization, an approximation scheme. We also get a useful local estimate of H and |∇u| that is independent of the size of u.

Recall that u is proper if each set {s ≤ u ≤ t} is compact. Write H_{+} := max(0, H_{∂E_{0}}).

3.1 Weak Existence Theorem. Let M be a complete, connected Riemannian n-manifold without boundary. Suppose there exists a proper, locally Lipschitz, weak subsolution of (††) with a precompact initial condition.

Then for any nonempty, precompact, smooth open set E_{0} in M, there exists a proper, locally Lipschitz solution u of (††) with initial condition E_{0}, which is unique on M \ E_{0}. Furthermore, the gradient of u satisfies the estimate

(3.1)  |∇u(x)| ≤ sup_{∂E_{0}∩B_{r}(x)} H_{+} + \frac{C(n)}{r},  a.e. x ∈ M \ E_{0},

for each 0 < r ≤ σ(x), where σ(x) > 0 is defined in Definition 3.3.

Remarks. 1. The subsolutions required by the Theorem will exist whenever M has some mild growth at infinity. In particular, if M is asymptotically flat, or more generally, asymptotically conic, then M will possess a subsolution of the form

v = C log |x|

in the asymptotic region.

2. As a consequence of (3.1), we can drop the local Lipschitz bounds in the Compactness Theorem in favor of adding a constant.

Let v be the given subsolution at infinity, F_{L} := {v < L}. We may assume that E_{0} ⊆ F_{0}. The domain Ω_{L} := F_{L} \ Ē_{0} is precompact.

To prove the theorem, we employ the following approximate equation, known as elliptic regularization (see [I1]).

(⋆)_{ε}  \{ \begin{matrix}E^{ε}u^{ε} := div (\frac{∇u^{ε}}{\sqrt{|∇u^{ε}|^{2} + ε^{2}}} ) − \sqrt{|∇u^{ε}|^{2} + ε^{2}} & = 0 & in Ω_{L}, \\ u^{ε} & = 0 & on ∂E_{0}, \\ u^{ε} & = L − 2 & on ∂F_{L}. \end{matrix}

The relation between ε and L will be revealed below.

This equation has a geometric interpretation: it states that the downward translating graph

N_{t}^{ε} := graph (\frac{u^{ε}(x)}{ε} − \frac{t}{ε}) , −∞ < t < ∞,

solves the inverse mean curvature flow (⋆) in the manifold M × R. To see this, define

(3.2) U^{ε}(x, z) := u^{ε}(x) − εz, (x, z) ∈ Ω_{L} × R,

so N_{t}^{ε} = {U^{ε} = t}. Then assuming smoothness, one checks that U^{ε} satisfies (⋆⋆) on Ω_{L}×R if and only if u^{ε} satisfies (⋆)_{ε} on Ω_{L}.

3.2 Example. Note that the natural units of ε are 1/x, since t is unitless. So by scaling ε → 0, we are actually scaling x → ∞. This suggests that our more difficult problems will be in the large rather than in the small.

In fact, for positive ε, solutions of (⋆)_{ε} do not exist globally, but only on a ball B_{c/ε}. The Maple-computed figure exhibits a rotationally symmetric solution u(x) = f(|x|) on R^{3} with ε = 1. Near x = 0, u ∼ (n − 1) log |x|. At |x| = c ≈ 1.98, u ceases to exist; its first and second derivatives diverge to ∞.

Downward Translating Solution.

A solution u^{ε} can be obtained from u by

u^{ε}(x) := u(εx), 0 < |x| < c/ε.

(Thus L cannot exceed c/ε in the flat case.) As ε → 0, then modulo vertical translations, u^{ε} converges to the solution

u(x) := (n − 1) log |x|, x ∈ R^{n} \ {0}, of (⋆⋆), corresponding to an expanding sphere.

Estimate of H. We will derive a local estimate for H for smooth solutions of (⋆). The −|A|^{2}/H term in (1.3) is so strong that the estimate is independent of the oscillation of u. The estimate improves on (1.4).

3.3 Definition. For any x ∈ M, define σ(x) ∈ (0, ∞] to be the supremum of radii r such that B_{r}(x) ⊂⊂ M,

Rc ≥ −\frac{1}{100nr^{2}} in B_{r}(x),

and there exists a C^{2} function p on B_{r}(x) such that

p(x) = 0,  p ≥ d_{x}^{2}  in B_{r}(x),

yet

|∇p| ≤ 3d_{x} and ∇^{2}p ≤ 3g on B_{r}(x),

where d_{x} is the distance to x. In flat space, p(y) := |y − x|^{2} does the job with r = ∞. More generally, a sufficient condition is the existence of a diffeomorphism of B_{r}(x) with a smooth ball in R^{n} such that

|g − δ| ≤ \frac{1}{100} and |g_{ij,k}| ≤ \frac{1}{100r} in B_{r}(x).

Then p(y) := 100|y − x|^{2}/99 will do the trick, where | · | represents distance in coordinates.

Now we will obtain an estimate of H using the maximum principle. Fix x and 0 < r < σ(x), and write B_{r} for B_{r}(x). Define the speed function ψ = 1/H and transform (1.3) to

(3.3)  \begin{aligned}\frac{∂ψ}{∂t} & = ψ^{2}(Δψ + |A|^{2}ψ + Rc(ν, ν)ψ) \\ & ≥ ψ^{2}Δψ + \frac{ψ}{n − 1} − \frac{ψ^{3}}{100nr^{2}} , \end{aligned}

on N_{t} ∩ B_{r}. We seek a function ϕ = ϕ(y) that vanishes on ∂B_{r} and is a subsolution of (3.3) along N_{t} ∩ B_{r}. For this purpose, it suffices that

(3.4)  \frac{∂ϕ}{∂t} ≤ ϕ^{2}Δϕ + \frac{ϕ}{2n},

provided we assume ϕ ≤ r.

We have the following relations to the ambient derivatives of ϕ,

\frac{∂ϕ}{∂t} = ψν · ∇ϕ, Δϕ = S: ∇^{2}ϕ − \frac{1}{ψ} ν · ∇ϕ,

where ∂ϕ/∂t is the derivative following a particle on N_{t}, S is the orthogonal projection from T_{x}M to T_{x}N_{t}, and ν is the exterior normal of E_{t}.

If we assume that ϕ < ψ initially on N_{t} ∩ B_{r}, then at the first point of contact, we have ϕ = ψ. Therefore, for the purpose of producing a subsolution, we may replace ψ by ϕ in the ambient relations and substitute the modified relations into (3.4), to obtain the following sufficient condition for ϕ = ϕ(y) to be a subsolution of (3.3),

(3.5)  0 ≤ ϕ^{2}S: ∇^{2}ϕ − 2ϕν · ∇ϕ + \frac{ϕ}{2n},

where this is required to hold for every x ∈ B_{r} and every hyperplane S and unit vector ν in T_{x}M. Define

ϕ(y) := \frac{A}{r}(r^{2} − p(y))_{+},

where p is as defined above. Then ϕ = 0 on ∂B_{r}, and ϕ ≤ r provided A ≤ 1. By the definition of p, |∇p| ≤ 3r and S: ∇^{2}p ≤ 3(n−1), so by calculation, ϕ satisfies (3.5) provided 3nA^{2} + 6A ≤ 1/2n, for which A ≤ 1/6n(n + 1) suffices. This shows that the function

h(x) := \frac{Br}{(r^{2} − p(y))_{+}}

is a smooth supersolution on N_{t} ∩ B_{r} of the evolution equation (1.3) for H wherever h is finite, provided B ≥ 6n(n + 1).

Fix t, assume x ∈ N_{t}, and define the parabolic boundary of the flow to be

P_{r} = P_{r}(x, t) := (B_{r} ∩ N_{0}) × {0} ∪ (∪_{0≤s≤t}(B_{r} ∩ ∂N_{s}) × {s})

and

H_{r} = H_{r}(x, t) := sup_{(y,s)∈P_{r}} H(y, s),

where we allow N_{s} to have a smooth boundary ∂N_{s}. Set

B = max(rH_{r}, 6n(n + 1)),

which ensures that h is a supersolution and h ≥ H on P_{r}. By the maximum principle, h ≥ H everywhere on N_{s} ∩ B_{r}, 0 ≤ s ≤ t, so in particular

H(x, t) ≤ h(x, t) = \frac{B}{r} = max (H_{r}, \frac{6n(n + 1)}{r}) .

We have proven the following lemma.

Interior Estimate of H. Let (N_{t})_{0≤s≤t} solve (⋆) smoothly in M, where N_{t} may have boundary. Then for each x ∈ N_{t} and each r < σ(x), we have

(3.6)  H(x, t) ≤ max (H_{r}, \frac{C(n)}{r}) ,

where H_{r} is the maximum of H on P_{r}, the parabolic boundary of the intersection of the flow with B_{r}(x), 0 ≤ s ≤ t.

To solve equation (⋆)_{ε}, we will estimate solutions of the following family of equations,

(⋆)_{ε,τ}  \{ \begin{matrix}E^{ε}u^{ε,τ} := div (\frac{∇u^{ε,τ}}{\sqrt{|∇u^{ε,τ} |^{2} + ε^{2}}} ) − \sqrt{|∇u^{ε,τ} |^{2} + ε^{2}} & = 0 & in Ω_{L}, \\ u & = 0 & on ∂E_{0}, \\ u & = τ & on ∂F_{L}, \end{matrix}

for 0 ≤ τ ≤ L − 2.

3.4 Lemma. Suppose the subsolution v provided in Theorem 3.1 is smooth, with ∇v ≠ 0. Then for every L > 0, there is ε(L) > 0 such that for 0 < ε ≤ ε(L) and 0 ≤ τ ≤ L − 2, a smooth solution of (⋆)_{ε,τ} on Ω̄_{L} satisfies the following estimates

u^{ε,τ} ≥ −ε in Ω̄_{L}, u^{ε,τ} ≥ v + τ − L in F̄_{L} \ F_{0}. (3.7) |∇u^{ε,τ} | ≤ H_{+} + ε on ∂E_{0}, |∇u^{ε,τ} | ≤ C(L) on ∂F_{L}, (3.8)

|∇u^{ε,τ} (x)| ≤ max_{∂Ω_{L}∩B_{r}(x)} |∇u^{ε,τ} | + ε + \frac{C(n)}{r}, x ∈ Ω̄_{L}, (3.9)

|u^{ε,τ} |_{C^{2,α}(Ω̄_{L})} ≤ C(ε, L), (3.10)

for any r with 0 < r ≤ σ(x). Here H_{+} = max(0, H_{∂E_{0}}).

Note that (3.8) and (3.9), with r = σ(x), yield uniform C^{0,1} estimates (independent of ε and L) on each compact set, for all sufficiently large L.

Proof of Lemma 3.4. 1. We will use supersolutions and subsolutions, (3.6), and standard theorems. Write u = u^{ε,τ}.

The particular size of Ω_{L} on which u exists is determined by the availability of subsolutions, as can be seen in Example 3.2. Subsolutions are delicate and require conditions on the asymptotic region, since the surfaces really would like to jump instantly to infinity.

First we construct a subsolution that bridges from E_{0} to where v starts. It is a perturbation of zero, and hence allows for unrestricted jumps in the compact part of the manifold. Define G_{0} = E_{0}, G_{s} := {x : dist(x, E_{0}) < s}. Select s_{L} so that G_{s_{L}} contains F_{L}. (This is possible since M is connected and E_{0} is nonempty.) Let Σ be the cut locus of E_{0} in M. On M \ E_{0} \ Σ, the distance function is smooth, each point is connected to E_{0} by a unique length-minimizing geodesic γ, and ∂G_{s} foliates a neighborhood of γ. Differentiating along γ, we have by (1.2),

\frac{∂H}{∂s} = −|A|^{2} − Rc(ν, ν) ≤ C_{1}  on ∂G_{s} \ Σ,  0 ≤ s ≤ s_{L},

where C_{1} = C_{1}(L), yielding

H_{∂G_{s}} ≤ max_{∂E_{0}} H_{+} + C_{1}s ≤ C_{2}  on ∂G_{s} \ Σ,  0 ≤ s ≤ s_{L},

where C_{2} = C_{2}(L). Now consider the prospective subsolution

v_{1}(x) := f(s) = f(dist(x, G)), x ∈ Ḡ_{s_{L}} \ E_{0},

with f′ < 0. Then

(g^{ij} − ν^{i}ν^{j}) ∇_{ij}^{2}v_{1} = f′H_{∂G_{s}} ≥ C_{2}f′,

and hence

\begin{aligned}\sqrt{(f′)^{2} + ε^{2}} E^{ε}v_{1} & = (g^{ij} − \frac{(f′)^{2}ν^{i}ν^{j}}{(f′)^{2} + ε^{2}} ) ∇_{ij}^{2}v_{1} − (f′)^{2} − ε^{2} \\ & ≥ C_{2}f′ + \frac{ε^{2}f′′}{(f′)^{2} + ε^{2}} − (f′)^{2} − ε^{2}, \end{aligned}

on the smooth part, G_{s_{L}} \ E_{0} \ Σ. Therefore, E_{2}^{ε}v_{1} ≥ 0 on this set, provided

((f′)^{2} + ε^{2})((f′)^{2} + ε^{2} − f′C_{2}) ≤ ε^{2}f′′.

Set

f(s) := \frac{ε}{A}(−1 + e^{−As}), 0 ≤ s ≤ s_{L}.

Then we have ε^{2} ≤ |f′| ≤ ε, provided that we impose ε ≤ ε(A, L) := e^{−As_{L}}. Choosing A = A(L) := 4 + 2C_{2}, we have

((f′)^{2} + ε^{2})((f′)^{2} + ε^{2} − f′C_{2}) ≤ 2ε^{2}(2ε^{2} + C_{2}|f′|) ≤ 2ε^{2}(2 + C_{2})|f′| = ε^{2}f′′,

as required. This shows that for sufficiently small ε, the function

v_{1}(x) = \frac{ε}{4 + 2C_{2}} (−1 + e^{−(4+2C_{2})s}), s = dist(x, ∂G_{0}),

is a smooth subsolution for E^{ε} on G_{s_{L}} \ E_{0} \ Σ.

We claim that v_{1} is a viscosity subsolution of E^{ε} on all of G_{s_{L}} \Ē_{0} (see [CIL]). Suppose ϕ is a smooth function tangent to v_{1} from above at a point x. Since f′ < 0, {ϕ = v_{1}(x)} is locally a smooth hypersurface tangent to G_{s} at x from outside, from which it follows by solving the Riccati equation backward along γ that x ∉ Σ. Therefore, we have E^{ε}ϕ(x) ≥ 0, which proves the claim.

Since u ≥ v_{1} on the boundary, it follows by the maximum principle for viscosity solutions that

(3.11)  u ≥ v_{1} ≥ −ε in Ω̄_{L},  \frac{∂u}{∂ν} ≥ −ε on ∂E_{0}.

2. Next, consider the function

v_{2} := \frac{L − 1}{L}v + τ − (L − 1).

Clearly E^{0}v_{2} > 0 on F̄_{L} \ F_{0}. Since the domain is compact, for all sufficiently small ε we obtain E^{ε}v_{2} > 0. Note that

u ≥ −ε ≥ v_{2} on ∂F_{0}, u = τ = v_{2} on ∂F_{L},

since 0 ≤ τ ≤ L − 2. Then by the maximum principle,

(3.12)  u ≥ v_{2} ≥ v + τ − L in F̄_{L} \ F_{0},  \frac{∂u}{∂ν} ≥ −C(L) on ∂F_{L},

Since any constant is a supersolution of (⋆)_{ε,τ}, we obtain

(3.13)  u ≤ τ in Ω̄_{L},  \frac{∂u}{∂ν} ≤ 0 on ∂F_{L}.

3. Next we construct a supersolution along ∂E_{0}. Choose a smooth function v_{3} vanishing on ∂E_{0}, such that

H_{+} < \frac{∂v_{3}}{∂ν} ≤ H_{+} + ε along ∂E_{0}.

This implies that for sufficiently small δ > 0, |∇v_{3}| > 0 and E^{0}v_{3} < 0 in the neighborhood U := {0 ≤ v_{3} ≤ δ}. Now define the sped-up function

v_{4} := \frac{v_{3}}{1 − v_{3}/δ} , x ∈ U.

Then E^{0}v_{4} < 0 as well, and v_{4} → ∞ on ∂U \ ∂E_{0}. Then for sufficiently small ε, depending on L, E^{ε}v_{4} < 0 on the set V := {0 ≤ v_{4} ≤ L}. Since u ≤ L − 2 by (3.13), we have u ≤ v_{4} on ∂V. Then by the maximum principle, we obtain u ≤ v_{4} on V, and therefore

(3.14)  \frac{∂u}{∂ν} ≤ \frac{∂v_{4}}{∂ν} = \frac{∂v_{3}}{∂ν} ≤ H_{+} + ε  on ∂E_{0},

for sufficiently small ε. Collecting together (3.11), (3.12), (3.13), and (3.14), we have proven (3.7) and (3.8).

4. Let N_{t}^{ε,τ} denote the level-set {U = t} of the function U(x, z) := u^{ε,τ} (x) − εz, −∞ < t < ∞. Equation (⋆)_{ε,τ} asserts

H_{N_{t}^{ε,τ}} = \sqrt{|∇u|^{2} + ε^{2}},

which says as pointed out above that N_{t}^{ε,τ} solves the inverse mean curvature flow in Ω_{L}×R. Let B̃ := B_{r}^{n+1}(x, z) be an (n+1)-dimensional ball in M ×R. Since the parabolic boundary of the flow N_{t}^{ε,τ} is nothing more than various translates of ∂Ω_{L} and |∇u| is independent of z, we may apply (3.6) to N_{t}^{ε,τ} in B̃ to yield

\sqrt{|∇u|^{2} + ε^{2}} ≤ sup_{t} max_{∂N_{t}^{ε,τ} ∩B̃} \sqrt{|∇u|^{2} + ε^{2}} + \frac{C(n)}{r}

≤ max_{∂Ω_{L}∩B_{r}(x)} |∇u| + ε + \frac{C(n)}{r},

for x ∈ M \ E_{0} and any r with 0 < r ≤ σ_{M}(x) = σ_{M×R}(x, u(x)/ε). This is (3.9).

Then (3.8) and (3.9) yield the estimate |u|_{C^{0,1}(Ω̄_{L})} ≤ C(L) for all sufficiently small ε. Following the method of [GT, Thm. 13.2], the Nash-Moser-De Giorgi estimates yield |u|_{C^{1,α}(Ω̄_{L})} ≤ C(ε, L), where α = α(Ω_{L}). The Schauder estimates [GT] complete the proof of (3.10).

3.5 Approximate Existence Lemma. Under the hypotheses of Lemma 3.4, a smooth solution of (⋆)_{ε} exists.

Proof. 1. We use the method of continuity applied to (⋆)_{ε,τ}, 0 ≤ τ ≤ L − 2. First let us prove that there is a solution for τ = 0 and small enough ε. Set u = εw and rewrite (⋆)_{ε,τ} as

F^{ε}(w) := div (\frac{∇w}{\sqrt{|∇w|^{2} + 1}}) − ε\sqrt{|∇w|^{2} + 1} = 0,

with w = 0 on ∂Ω_{L}. Clearly the map

F : C_{0}^{2,α}(Ω̄_{L}) × R → C^{α}(Ω̄_{L}),

defined by F (w, ε) := F^{ε}(w) is C^{1}, and possesses the solution F^{0}(0) = 0. The linearization of F^{0} at w = 0 is given by

DF^{0}|_{0} = Δ : C_{0}^{2,α}(Ω̄_{L}) → C^{α}(Ω̄_{L}),

the ordinary Laplace-Beltrami operator, an isomorphism. Then by the Implicit Function Theorem there is a solution of F^{ε}(w) = 0 for sufficiently small ε, and hence of (⋆)_{ε,τ} with τ = 0.

2. Next we fix ε and vary τ. Let I be the set of τ such that (⋆)_{ε,τ} possesses a solution. Now I contains 0 by Step 1, and I ∩ [0, L − 2] is closed by estimate (3.10) and the Arzela-Ascoli theorem.

Let us prove that I is open. Let π be the boundary values map u ↦ u|∂Ω, and define G^{τ} (u) := (E^{ε}(u), π(u) − τχ_{∂F_{L}}), so that (⋆)_{ε,τ} is equivalent to G^{τ} (u) = (0, 0). Clearly the map

F : C^{2,α}(Ω̄_{L}) × R → C^{α}(Ω̄_{L}) × C^{2,α}(∂Ω_{L}),

defined by F (u, τ) = G^{τ} (u), is C^{1}. The linearization of G^{τ} at a solution u is given by

DG^{τ} |_{u} = ( \begin{matrix}DE^{ε}|_{u} \\ π \end{matrix} ) : C^{2,α}(Ω̄_{L}) → C^{α}(Ω̄_{L}) × C^{2,α}(∂Ω_{L}).

Now E^{ε} has the form

E^{ε}(u) = ∇_{i}A^{i}(∇u) + B(∇u),

which is independent of u, so it follows by the maximum principle that the linearization

D^{ε}E|_{u}(v) = ∇_{i}(A_{p_{j}}^{i}(∇u)∇_{j}v) + B_{p_{j}}(∇u)∇_{j}v

possesses only the zero solution. Then using existence theory and Schauder estimates up to the boundary, DG^{τ} |_{u} is seen to be an isomorphism.

Then by the Implicit Function Theorem, the set of τ for which G^{τ} (u) = (0, 0) is solvable (namely I) is open. Therefore L − 2 ∈ I, which proves existence of u^{ε} in C^{2,α}. Smoothness follows by Schauder estimates.

Now we are prepared to prove the existence of the exact solution u. Because the downward translating graphs are an exact solution of (⋆) one dimension higher, we may use the Compactness Theorem as it stands to pass these sets to limits, obtaining a family of cylinders in M × R, which we will slice by M to obtain a family of surfaces weakly solving (⋆).

Proof of Theorem 3.1. 1. First assume that v is smooth with nonvanishing gradient. By Lemma 3.5, for each L > 0 there exists a smooth solution u^{ε} = u^{ε,L−2} of (⋆)_{ε} on Ω_{L}, where ε = ε(L) → 0 as L → ∞. Combining (3.8) and (3.9), we find that

|∇u^{ε}(x)| ≤ max_{∂E_{0}∩B_{r}(x)} H_{+} + 2ε + \frac{C(n)}{r},

on each compact subset of M \ E_{0} and sufficiently large L. By the Arzela-Ascoli theorem, there exists L_{i} → ∞, ε_{i} → 0, a subsequence u_{i}, and a locally Lipschitz function u such that

u_{i} → u,

locally uniformly on M \ E_{0}, and u satisfies estimate (3.1) in the limit. By (3.7) with τ = L − 2,

u ≥ 0 in M \ E_{0}, u → ∞ as x → ∞.

Let U_{i}(x, z) = u_{i}(x) − ε_{i}z as in (3.2), and set U(x, z) := u(x). Then

U_{i} → U,

locally uniformly on (M \ E_{0}) × R with local Lipschitz bounds. The sets

N_{t}^{i} := \{U_{i} = t\} = graph (\frac{u_{i}}{ε_{i}} − \frac{t}{ε_{i}} ) , −∞ < t < ∞,

smoothly solve (⋆) with H > 0 in the region Ω_{L_{i}} × R, so Lemma 2.3 implies that U_{i} satisfies the variational formulation (1.5) on Ω_{L_{i}} × R. Then the Compactness Theorem 2.1 implies that U satisfies (1.5) on (M \ Ē_{0}) × R.

Let us check that u satisfies (1.5) on M \ Ē_{0}. Let v be a locally Lipschitz function such that {v ≠ u} ⊂⊂ M \ Ē_{0}. Let ϕ(z) be a cutoff function with |ϕ_{z}| ≤ 1, ϕ = 1 on [0, S] and ϕ = 0 on R \ (−1, S + 1). Inserting V (x, z) := ϕ(z)v(x) into J_{U}(U) ≤ J_{U}(V), we obtain

∫_{K×[−1,S+1]} |∇u| + u|∇u| dx dz ≤ ∫_{K×[−1,S+1]} ϕ|∇v| + v|ϕ_{z}| + ϕv|∇u| dx dz,

where K contains {u ≠ v}. Dividing by S and passing S → ∞ proves that u satisfies (1.5). Finally, extend u negatively to E_{0} so that E_{0} = {u < 0}. This completes the proof of existence of the initial value problem (††) in the event that v is smooth with ∇v ≠ 0.

2. In the general case, fix L > 0 and select an open set U_{L} with smooth boundary such that F_{L} ⊂⊂ U_{L} ⊂⊂ M. Modify the metric on U_{L} near ∂U_{L} to obtain a complete metric g_{L} on U_{L} such that g_{L} = g on F_{L}, g_{L} ≥ g on all of U_{L}, and near ∂U_{L}, g_{L} is isometric to a Riemannian cone of the form

g_{L} = Cs^{2}g_{∂U_{L}} + ds^{2} in ∂U_{L} × [C, ∞).

For some α > 0, α log s is a smooth subsolution of (1.5) on ∂U_{L} × [C, ∞). By step 1, there exists a proper solution u_{L} of (††) in (M_{L}, g_{L}) with initial condition E_{0}.

By the reasoning of (1.18), min(v, L) is a weak subsolution of (⋆⋆) in U_{L} with respect to g. Since g = g_{L} where v < L and otherwise g_{L} ≥ g, it follows that min(v, L) remains a weak subsolution when considered with respect to the metric g_{L}. Then by the Comparison Theorem 2.2(ii), we have u_{L} ≥ v on F_{L}. Passing L → ∞ and taking a convergent subsequence via (3.1), the Compactness Theorem 2.1 yields a solution u defined everywhere on M, with u ≥ v. Theorem 2.2(iii) implies that u is unique. The gradient bound (3.1) is preserved.

Remarks. 1. By passing all of the surfaces N_{t}^{i} to limits, we can show that at a jump, the interior U of {u = t} × R is foliated by surfaces that minimize area subject to an obstacle condition consisting of the vertical walls ∂E_{t} and ∂E_{t}^{+}. Each such surface is either a vertical cylinder or a smooth graph over an open subset of U. This construction yields a high-speed movie of the jump from N_{t} to N_{t}^{+}.

2. We mention an alternate weak formulation of the inverse mean curvature flow. We call the locally Lipschitz function u a solution if there exists a measurable vector field ν such that

(3.15)

|ν| ≤ 1, ∇u · ν = |∇u| a.e.,
∫_{Ω} ∇ϕ · ν + ϕ|∇u| = 0 for all ϕ ∈ C_{c}^{1}(Ω).

The vector field ν extends ∇u/|∇u| as a calibration across the gulfs. In fact, ν is the projection to T M of the normal vector of the foliation constructed above. The following properties hold.

(i) Formulation (3.15) implies the variational formulation (1.5).

(ii) Compactness holds for (3.15), and consequently the existence procedure does too.

(iii) Uniqueness then implies that (3.15) is equivalent to (1.5) under the hypotheses of Theorem 3.1.

§4 Topological Consequences

We review the existence and topology of exterior regions, and prove that in an exterior region, a surface evolving weakly by inverse mean curvature flow remains connected. This sets the stage for bounding the Euler characteristic in §5.

Exterior and Trapped Regions. Let us recall some definitions and well-known results; see [G2,Hm,MSY,MY,SSY,Ga]. Let M be a complete 3-manifold with asymptotically flat ends. We allow M to have a smooth, compact boundary consisting of minimal surfaces (that is, surfaces of vanishing mean curvature).

Let K_{1} be the closure of the union of the images of all smooth, compact, immersed minimal surfaces in M. Since the region near infinity is foliated by spheres of positive mean curvature, K_{1} is compact.

The trapped region K is defined to be the union of K_{1} together with the bounded components of M \ K_{1}. The set K is compact as well. By minimization, there exists a smooth, embedded, stable surface separating any two ends of M, so M \K contains exactly one connected component corresponding to each end of M. The following lemma tells us about the smoothness and topology of these components.

4.1 Lemma. (i) The topological boundary of the trapped set K consists of smooth, embedded minimal 2-spheres. The metric completion, M′, of any connected component of M \ K is an “exterior region”, that is, M′ is connected and asymptotically flat, has a compact, minimal boundary, and contains no other compact minimal surfaces (even immersed).

(ii) A 3-dimensional exterior region M′ is diffeomorphic to R^{3} minus a finite number of open 3-balls with disjoint closures. The boundary of M′ minimizes area in its homology class.

Note that there is no curvature hypothesis. We take the metric completion rather than the closure because some component of K might be a nonseparating surface. The Lemma embodies various well-known theorems in the literature [MSY,MY,Ga], but we sketch the proof for completeness.

Proof. (i) First we show that K can be generated by a tamer collection of surfaces. Let C be the perimeter of some bounded open set containing K, and let A be the collection of smooth, compact, embedded minimal hypersurfaces in M such that

(a) |N| ≤ C, and

(b) the double cover of N that orients its normal bundle is a stable minimal immersion. (If N is transversely orientable, this says that N itself is stable.)

Let P be one of the surfaces used in defining K_{1}, that is, an arbitrary smooth, connected, compact, immersed minimal surface in M. Suppose P ∉ A. Let M̂ be the metric completion of M \ P, and let P̂ be the union of the (one or two) boundary components that correspond to P. Let N minimize area among smooth surfaces in M̂ that separate P̂ from infinity. Note that |N| ≤ C.

Since P ∉ A, either P has a segment of transverse double points, |P | > C, or P fails the stability (b). It follows that N cannot contain any component of P̂. By the strong maximum principle, N lies in M \ P. It follows that N ∈ A.

Therefore, either P ∈ A, or P is separated from infinity by a disjoint surface N ∈ A. This shows we may restrict to the class A in defining K.

Now ∂K is contained in the closure of the union of A. But the class A is compact in the C^{2} topology (see [SSY]), so in fact, each point of ∂K lies in some N in A.

Let N, N_{1} be distinct, connected surfaces in A that meet ∂K. We claim that they are disjoint. For otherwise, by the strong maximum principle, they cross, and using N ∪ N_{1} as a barrier for an area minimization problem, we find that N ∪ N_{1} is contained in int K, a contradiction.

Next the surfaces in A that meet ∂K are actually isolated from one another. Suppose we have connected surfaces N, N_{i} ∈ A such that each N_{i} is disjoint from N, but the C^{2} limit surface touches N. Then by the strong maximum principle, the limit surface equals N, so for sufficiently large i, N_{i} is either a single or double cover over N under the nearest point projection. In either case, the region between N_{i} and N is part of the interior of K, so that eventually N_{i} does not meet ∂K. This shows that a surface N ∈ A that meets ∂K is isolated from the other surfaces in A that meet ∂K.

It follows that ∂K consists of a finite union of disjoint, connected, stable minimal hypersurfaces. Each surface is either a component of K, or bounds K on one side. This proves the smoothness of ∂K. The spherical topology comes from (ii). The claims about M′ are immediate.

(ii) Let M′ be an exterior region. If M′ is not simply connected or has more than one end, then its universal cover must have several ends, which as above, are separated by some smooth, compact minimal surface. Clearly this surface is not contained entirely in the boundary, so its projection to M′ would violate the exterior region hypothesis. Therefore M′ is simply connected, with one end.

By the Loop Theorem [He,MY] and simple connectedness, ∂M′ consists of 2-spheres. By simple connectedness, M′ contains no one-sided RP^{2}, so by [MSY], any fake 3-cell in M′ would be surrounded by a least area sphere, which would contradict the exterior region hypothesis. The manifold obtained by filling in the boundary spheres with balls and compactifying at infinity is therefore a simply connected 3-manifold without fake 3-cell, hence is S^{3}. This implies that M′ has the topology claimed.

To see that ∂M′ is area-minimizing: using large spheres as barriers, there is some areaminimizing compact surface N homologous to ∂M′, but since M′ is an exterior region, N must lie in ∂M′, and therefore N = ∂M′.

Remark. Statement (i) can alternately be proven by flowing a large celestial 2-sphere forever by mean curvature. The mean curvature remains positive, so the flow is unidirectional, and the maximum principle keeps the spheres from the different ends from running into each other. The flowing surface can split into further spheres which eventually stabilize at a finite union of stable, smooth 2-spheres. The area swept out is free of minimal surfaces. This argument can be made rigorous by recent techniques of White [Wh].

The following lemma implies that a connected surface evolving in an exterior region remains connected. We are grateful to P. Li for clarifying the role of topology in part (ii) of the lemma.

4.2 Connectedness Lemma.

(i) A solution u of (1.5) has no strict local maxima or minima.

(ii) Suppose M is connected and simply connected with no boundary and a single, asymptotically flat end, and (E_{t})_{t>0} is a solution of (†) with initial condition E_{0}. If ∂E_{0} is connected, then N_{t} remains connected as long as it stays compact.

Proof. (i) If u possesses a strict local maximum or minimum, then there is an “island” E ⊂⊂ Ω, that is, a connected, precompact component of {u > t} or of {u < t} for some t. Define the Lipschitz function v by v = u on Ω \ E, v = t on E. Then (1.5) yields

∫_{E} |∇u| + u|∇u| ≤ ∫_{E} t|∇u|.

In the case of {u > t}, this immediately implies that u = t on E, a contradiction. In the case of {u < t}, we select t to ensure that u > t−1 on E, and again obtain a contradiction.

(ii) Let Ω := M \ Ē_{0}. Let t > 0. By (1.10), each N_{t} is approximated in C^{1} by earlier surfaces N_{s_{i}} at which there are no jumps. Therefore we may assume that N_{t} = {u = t}.

Let W := {u < t}. Because u is proper, W ∩ Ω is bounded. But by (i), no component of W can be compactly contained in Ω. Therefore, each component of W meets ∂Ω = ∂E_{0}. Since ∂E_{0} is connected, it follows that W is connected.

Let X := {u > t}. Since u is proper, X contains a neighborhood of infinity, which is connected since M has only one end. On the other hand, by (i), X has no precompact component. Therefore X is connected.

If N_{t} has more than one component, then by the connectedness of X and W, there is a closed loop γ that starts near ∂E_{0}, crosses N_{t} via the first component, and returns via the second component. By intersection theory, γ cannot be homotoped to zero, a contradiction to the simple connectedness.

§5 Geroch Monotonicity

The Geroch Monotonicity Formula states that the Hawking quasi-local mass of a connected surface in a manifold of nonnegative scalar curvature is monotone during the inverse mean curvature flow. It was used by Geroch [G] and Jang-Wald [JW] to argue for the Penrose Inequality in case the flow remains smooth forever. In this section, we establish the formula for the weak flow. An advantage of the elliptic regularization approximation is that it preserves most of the geometry, so a computation at the ε-level is feasible, which may then be passed to limits.

Monotonicity Calculation. Let us first demonstrate the monotonicity in the smooth case. Recall that the Hawking mass is defined by m_{H}(N) := |N|^{1/2}(16π−∫_{N} H^{2})/(16π)^{3/2}. We find by (1.1) and (1.3),

\frac{d}{dt} ∫_{N_{t}} H^{2} = ∫_{N_{t}} −2HΔ (\frac{1}{H} ) − 2|A|^{2} − 2Rc(ν, ν) + H^{2}.

The Gauss equation implies

K = K_{12} + λ_{1}λ_{2} = \frac{R}{2} − Rc(ν, ν) + \frac{1}{2} (H^{2} − |A|^{2})

where K is the Gauss curvature of N, K_{12} is the sectional curvature of M in the direction of T_{x}M, λ_{1}, λ_{2} are the principal curvatures of N in M, and R is the scalar curvature of M. Eliminating Rc in favor of R, we find

\frac{d}{dt} ∫_{N_{t}} H^{2} = ∫_{N_{t}} −2\frac{|DH|^{2}}{H^{2}} − |A|^{2} − R + 2K

= 4πχ(N_{t}) + ∫_{N_{t}} −2\frac{|DH|^{2}}{H^{2}} − \frac{1}{2}H^{2} − \frac{1}{2}(λ_{1} − λ_{2})^{2} − R

≤ \frac{1}{2} (16π − ∫_{N_{t}} H^{2})

provided that N_{t} is connected and R ≥ 0. (A similar calculation is used in studying positive scalar curvature via stable minimal surfaces, including the Schoen-Yau proof of the positive mass theorem.)

It follows that the quantity

e^{t/2}(16π − ∫_{N_{t}} H^{2})

is nondecreasing. Since |N_{t}|^{1/2} equals e^{t/2} up to a factor, this shows that m_{H}(N_{t}) is nondecreasing.

The estimate of ∫_{N_{t}} H^{2} is quite interesting in itself because it is independent of the detailed geometry of M except the scalar curvature.

Heuristically, when the surface jumps, we should obtain by Lemma 1.4(iv) and (1.15),

∫_{∂E_{t}^{+}} H^{2} ≤ ∫_{∂E_{t}} H^{2}, |∂E_{t}^{+}| = |∂E_{t}|,

at least for t > 0, which implies that the monotonicity is preserved even at a jump. This reasoning is valid at t = 0 as well, provided E_{0} is a minimizing hull. This is how the equation divines that there are no other minimal surfaces in M. In the remainder of this section, we substantiate this discussion.

Monotonicity Calculation, ε-Version. We will now reproduce the Geroch formula in a weak setting. Assume for now that N_{0} is smooth and M possesses a smooth subsolution at infinity. Let u^{ε} := u^{ε,L−2} be an approximator defined over Ω_{L} as in the proof of Theorem 3.1. Recall that N_{t}^{ε} := graph(u^{ε}/ε − t/ε), −∞ < t < ∞, is a smooth inverse mean curvature flow. Let ν be the downward unit normal to N_{t}^{ε}. Note H = −H⃗ · ν > 0.

Select a cutoff function ϕ ∈ C_{c}^{2}(R) such that ϕ ≥ 0, spt ϕ ⊆ [1, 5], and ∫ ϕ(z) dz = 1. Fix an arbitrary T > 0 and require L ≥ T +7, ε ≤ 1, so that ∂N_{t}^{ε} is disjoint from M ×spt ϕ for 0 ≤ t ≤ T. The boundary term disappears and we calculate

(5.1)  \begin{aligned}\frac{d}{dt} ∫_{N_{t}^{ε}} ϕH^{2} & = ∫_{N_{t}^{ε}} 2ϕH \frac{∂H}{∂t} + H^{2}∇ϕ · \frac{ν}{H} + ϕH^{2} \\ & = ∫_{N_{t}^{ε}} ϕ (−2HΔ (\frac{1}{H} ) − 2|A|^{2} − 2Rc(ν, ν)) + ∇ϕ · νH + ϕH^{2} \\ & = ∫_{N_{t}^{ε}} ϕ (−2\frac{|DH|^{2}}{H^{2}} − 2|A|^{2} − 2Rc(ν, ν) + H^{2}) \\ & − 2Dϕ · \frac{DH}{H} + ∇ϕ · νH, \end{aligned}

for 0 ≤ t ≤ T. In integrated form this becomes

(5.2) ∫_{N_{r}^{ε}} ϕH^{2} = ∫_{N_{s}^{ε}} ϕH^{2} + ∫_{r}^{s} ∫_{N_{t}^{ε}} ϕ (2\frac{|DH|^{2}}{H^{2}} + 2|A|^{2} + 2Rc(ν, ν) − H^{2})

+ 2Dϕ · \frac{DH}{H} − ∇ϕ · νH,

for 0 ≤ r < s ≤ T.

Estimates. Let us estimate each of these terms in turn, with an eye to making the above formula converge as ε → 0, L → ∞. Fix T > 0 and assume 0 ≤ t ≤ T. Our constants may depend on T but not on ε or L.

By (3.7), there is R(T) > 0 depending only on the subsolution v such that

(5.3)  N_{t}^{ε} ∩ (M × spt ϕ) ⊆ K(T) := (B_{R(T)} \ E_{0}) × [1, 5],  0 ≤ t ≤ T,

a fixed compact set. Applying the minimizing hull property to the supergraph E_{t}^{ε} compared to the perturbation E_{t}^{ε} ∪ K(T), we obtain the area estimate

(5.4)  |N_{t}^{ε} ∩ (M × spt ϕ)| ≤ |∂K(T)| = C(T),  0 ≤ t ≤ T.

From (3.6), (3.8), the fact that H_{N_{t}^{ε}} = |∇u^{ε}|, and (5.3), we have for L ≥ T + 8,

(5.5)  |H| ≤ C(T)  on N_{t}^{ε} ∩ (M × spt ϕ),  0 ≤ t ≤ T.

These two inequalities imply

(5.6)  ∫_{N_{t}^{ε}} ϕH^{2} + |∇ϕ · νH| ≤ C(T),  0 ≤ t ≤ T.

Next,

|2Dϕ · \frac{DH}{H} | ≤ \frac{|Dϕ|^{2}}{ϕ} + ϕ\frac{|DH|^{2}}{H^{2}} ≤ C + ϕ\frac{|DH|^{2}}{H^{2}} ,

since ϕ is C^{2} of compact support. Putting these two inequalities into (5.1), we obtain

(5.7)  \frac{d}{dt} ∫_{N_{t}^{ε}} ϕH^{2} ≤ ∫_{N_{t}^{ε}} −ϕ (\frac{|DH|^{2}}{H^{2}} + 2|A|^{2}) + C(T),  0 ≤ t ≤ T,

which implies together with (5.6) and (5.5) that

(5.8)  ∫_{0}^{T} ∫_{N_{t}^{ε}∩(M×[2,4])} \frac{|DH|^{2}}{H^{2}} + |DH|^{2} + |A|^{2} ≤ C(T),

using a ϕ such that ϕ = 1 on [2, 4]. For any sequence ε_{i} → 0, Fatou’s Lemma yields

(5.9)  lim inf_{i→∞} ∫_{N_{t}^{ε_{i}} ∩(M×[2,4])} \frac{|DH|^{2}}{H^{2}} + |DH|^{2} + |A|^{2} < ∞,  a.e. t ≥ 0.

Henceforth we shrink ϕ so that spt ϕ ⊆ [2, 4].

Convergence. As in the proof of Theorem 3.1, there are subsequences ε_{i} → 0, L_{i} → ∞, N_{t}^{i} = N_{t}^{ε_{i}} such that

(5.10)  N_{t}^{i} → Ñ_{t} = N_{t} × R locally in C^{1},  a.e. t ≥ 0,

where N_{t} = ∂E_{t} and (E_{t})_{t>0} is the unique solution of (†) for E_{0}.

We wish to pass (5.2) to limits. Let us first address the ∫_{N_{t}^{i}} ϕH^{2} term. By (1.14), this expression is lower semicontinuous as i → ∞ (at least where there is no jump). To make this an equality for a.e. t ≥ 0, we use growth control and Rellich’s theorem. By (5.7), the function ∫_{N_{t}^{i}} ϕH^{2} − C(T)t is monotone, 0 ≤ t ≤ T. Therefore by choosing a diagonal subsequence (labelled the same) we may arrange that

(5.11)  lim_{i→∞} ∫_{N_{t}^{i}} ϕH^{2} exists,  a.e. t ≥ 0.

Suppose t is such that (5.9) and (5.10) hold. It is possible to write the converging surfaces N_{t}^{i} simultaneously as graphs of C^{1} functions w_{i} over some smooth surface W, that is,

N_{t}^{i} ∩ (M × [2, 4]) = {x + w_{i}(x)ν_{W} (x) : x ∈ W } ∩ (M × [2, 4]), w_{i} ∈ C_{loc}^{1}(W),

and w_{i} → w locally in C^{1}, where Ñ_{t} is the graph of w. (Here addition indicates the normal exponential map.) Then by Rellich’s theorem, (1.13), and (5.9), there is a subsequence i_{j} such that

(5.12)  H_{N_{t}^{i_{j}}} → HÑ_{t}  in L^{2}(W ∩ (M × [2, 4]),

where the curvature function are projected to W for comparison. We conclude by (5.11) that the full sequence converges, namely

(5.13)  ∫_{N_{t}^{i}} ϕH^{2} → ∫Ñ_{t} ϕH^{2},  a.e. t ≥ 0.

Together with (5.6) and the bounded convergence theorem, this implies that

(5.14)  ∫_{r}^{s} ∫_{N_{t}^{i}} ϕH^{2} → ∫_{r}^{s} ∫Ñ_{t} ϕH^{2}.

for any 0 ≤ r < s.

Next we turn to the ∇ϕ · νH term. By (5.5) and (5.4), we have

∫_{N_{t}^{i}} |∇ϕ · νH| ≤ C(T) sup_{N_{t}^{i}} |∇ϕ · ν| → 0, a.e. t ≥ 0.

We have used the fact that N_{t}^{i} converges locally in C^{1} to the vertical cylinder Ñ_{t}, on which ν is perpendicular to ∇ϕ. Then by (5.6) and the bounded convergence theorem, we obtain

(5.15)  ∫_{r}^{s} ∫_{N_{t}^{i}} |∇ϕ · νH| → 0.
Next we address the |DH|^{2}/H^{2} term.

5.1 Lemma. For a.e. t ≥ 0,

H > 0 H^{n−1}-a.e. on N_{t}.

Proof. Clearly |∇u| > 0 a.e. with respect to the measure |∇u| dx. By the co-area formula, this implies that |∇u| exists and |∇u| > 0 for a.e. t ≥ 0 and H^{n−1}-a.e. x ∈ N_{t}. By (1.12), this implies the claimed result.

In particular the integral ∫Ñ_{t} |DH|^{2}/H^{2} makes sense for a.e. t.

5.2 Lower Semicontinuity Lemma. For each 0 ≤ r < s,

∫_{r}^{s} ∫Ñ_{t} ϕ\frac{|DH|^{2}}{H^{2}} ≤ lim inf_{i→∞} ∫_{r}^{s} ∫_{N_{t}^{i}} ϕ\frac{|DH|^{2}}{H^{2}} .

Proof. By (5.9), for a.e. t ≥ 0 there is a subsequence i_{j} such that

(5.16)  sup_{j} ∫_{N_{t}^{i_{j}} ∩(M×[2,4])} \frac{|DH|^{2}}{H^{2}} < ∞.

Let N̂ be a connected component of Ñ_{t}∩(M ×[2, 4]), and let N̂^{j} be a connected component of N_{t}^{i_{j}} ∩ (M × [2, 4]) converging locally in C^{1} to N̂.

Let a_{j} be the median of log H on N̂^{j}. By (5.16) and the Rellich theorem, there exists a ∈ [−∞, ∞), f ∈ L^{2}(N̂), and a further subsequence such that a_{j} → a and

log HN̂^{j} − a_{j} → f in L^{2}(W) and a.e. on W,

where N̂^{j}, N̂ are written as C^{1} graphs over some nearby smooth surface W, and the functions are projected to W.

If a = −∞, then log HN̂^{j} → f + a = −∞ a.e., so HN̂^{j} → 0 a.e. on W. By the boundedness of H and the weak convergence (1.13), it follows that HN̂ = 0 a.e. on N̂. But by Lemma 5.1, this case is excluded for a.e. t ≥ 0.

Therefore we assume a > −∞. Then

log HN̂^{j} − a_{j} → log HN̂ − a in L^{2}(W).

It follows that we have weak convergence of DH/H in each chart W. Then by the usual lower semicontinuity, we obtain for a.e. t ≥ 0,

∫Ñ_{t} ϕ\frac{|DH|^{2}}{H^{2}} ≤ lim inf_{i→∞} ∫_{N_{t}^{i}} ϕ\frac{|DH|^{2}}{H^{2}} ,

which implies the desired result by Fatou’s Lemma.

The final term of (5.2) converges only weakly to zero.

5.3 Lemma. For each 0 ≤ r < s,

∫_{r}^{s} ∫_{N_{t}^{i}} Dϕ · \frac{DH}{H} → 0.

Proof. As is (5.15), the idea is that Dϕ and DH are orthogonal in the cylindrical limit. From the previous lemma, DH/H converges weakly subsequentially for individual times, but it is tricky to control the time direction. To do this, we recognize the inner integral as a time derivative, using the fact that N_{t}^{i} is moving rigidly by translation. Defining

g_{i}(t) := ∫_{N_{t}^{i}} −ε_{i}ϕ(z)\frac{∂}{∂z} · \frac{DH}{H} dμ_{N_{t}^{i}}(x, z)

= ∫_{N_{0}^{ε_{i}}} −ε_{i}ϕ(z − t/ε_{i})\frac{∂}{∂z} · \frac{DH}{H} dμ_{N_{0}^{ε_{i}}} (x, z),

we notice that

\frac{d}{dt}g_{i}(t) = ∫_{N_{0}^{ε_{i}}} ϕ′(z − t/ε_{i})\frac{∂}{∂z} · \frac{DH}{H}

= ∫_{N_{t}^{i}} ϕ′(z)\frac{∂}{∂z} · \frac{DH}{H}

= ∫_{N_{t}^{i}} Dϕ · \frac{DH}{H}

=: f_{i}(t).

We have by Cauchy’s inequality, (5.4), and (5.8),

sup_{i} ∫_{0}^{T} |f_{i}| < ∞, ∫_{0}^{T} |g_{i}| ≤ C(T)ε_{i} → 0,

for each T > 0, hence

f_{i} ⇀ 0 on [0, ∞),

in the sense of measures, which gives the result.

Weak Second Fundamental Form. In order to cope with the |A|^{2} term, we define the second fundamental form of a surface in W^{2,2} ∩ C^{1}. See Hutchinson [Hut] for a related definition. Let N be a C^{1} hypersurface of an ambient manifold M with induced metric h = (h_{ij}), orthogonal projection (h_{i}^{j}) from T M to T N, unit normal (ν^{i}), and weak mean curvature H in L_{loc}^{1}(N), as defined by the first variation formula (1.11).

Suppose p = (p_{ij}) is a smooth, compactly supported symmetric 2-tensor defined on M, and apply the first variation formula to the vector Y^{j} = g^{jl}p_{lk}ν^{k} to derive in the smooth case

(5.17)  ∫_{N} Hν^{j}p_{jk}ν^{k} = ∫_{N} h^{ij}∇_{i}(p_{jk}ν^{k}) = ∫_{N} h^{ij}∇_{i}p_{jk}ν^{k} + h^{ij}p_{jk}h^{kl}A_{li},

recalling that A_{ij} = h_{i}^{k}∇_{k}ν^{l}h_{lj}.

If N is C^{1} and H exists as a locally integrable function, we call a locally integrable section A = (A_{ij}) of Sym^{2}(T^{∗}N) the second fundamental form if the far right and left sides of (5.17) are equal for every p in C_{c}^{1}(Sym^{2}(T^{∗}M)).

Assuming N is C^{1}, it can be verified that |A| ∈ L_{loc}^{2}(N) if and only if N can be written locally in coordinates as the graph of a W^{2,2} function w. (The tensor A_{ij} can be written as the Hessian of w, with a correction that depends only on the normal vector and the Christoffel symbols of g. See (7.10).) Inserting p_{ij} = ϕg_{ij}, we find that if A exists, then indeed H = h^{ij}A_{ij} a.e., where H is defined by (1.11).

It follows from weak compactness, Riesz Representation, and (1.13) that if N^{i} → N locally in C^{1} and

sup_{i} ∫_{N^{i}} |A_{N^{i}}|^{2} < ∞,

then A_{N} exists in L^{2}(N) with the weak convergence

∫_{N^{i}} p^{jk}A_{jk}^{N^{i}} → ∫_{N} p^{jk}A_{jk}^{N}, p ∈ C_{c}^{0}(Sym^{2}(T M)),

and lower semicontinuity

(5.18)  ∫_{N} |A_{N}|^{2} ≤ lim inf_{i→∞} ∫_{N_{i}} |A_{N_{i}}|^{2}.

On the other hand, if a C^{1} surface N is given satisfying ∫_{N} |A|^{2} < ∞, and we are allowed to choose N_{i} by mollification, we may arrange that each N_{i} is smooth and

(5.19)  N_{i} → N  strongly in C^{1} and W^{2,2},

in the sense that N_{i}, N are represented locally as graphs of functions w_{i}, w such that w^{i} → w in C^{1} ∩ W^{2,2}. (We can even arrange that N_{i} approaches N from one side.)

In this situation, let λ_{1}, λ_{2} be the eigenvalues of A with respect to h and consider the function in L^{1}(N) defined by

K := K_{12} + λ_{1}λ_{2},

where K_{12} is the sectional curvature function of the ambient manifold evaluated on T_{x}N. The following lemma is now a consequence of the Gauss and Gauss-Bonnet formulae applied to the approximators N_{i}.

5.4 Weak Gauss-Bonnet Formula. Suppose N is a compact C^{1} surface in a 3-manifold, satisfying ∫_{N} |A|^{2} < ∞. Then

∫_{N} K_{12} + λ_{1}λ_{2} = 2πχ(N),

where χ denotes the Euler characteristic.

From this we have the following.

5.5 W^{2,2} Lemma. Let N_{t} be the limiting surfaces defined in (5.10). Then

∫_{N_{t}} |A|^{2} ≤ C(T), 0 ≤ t ≤ T.

Proof. By (5.9) and (5.18),

∫_{N_{t}} |A|^{2} < ∞ for a.e. t ≥ 0.

Since the surfaces N_{t} are compact with locally uniform C^{1} estimates, χ(N_{t}), |N_{t}|, and sup_{N_{t}} |K_{12}| are bounded for 0 ≤ t ≤ T. By Lemma 5.4, this yields

∫_{N_{t}} λ_{1}λ_{2} ≤ C(T), a.e. t ∈ [0, T].

Since H = λ_{1} + λ_{2} is bounded, this yields the desired estimate for a.e. t ∈ [0, T]. It follows for all t by (5.18).

Remark. A similar result can be proven for inverse mean curvature flow in dimensions n < 8, by a local version of the Gauss-Bonnet formula suitable for graphs.

We need one more technical lemma.

5.6 Lemma. Suppose E is precompact, E′ = E, and ∂E is C^{1,1}. Then either ∂E is a smooth minimal surface, or ∂E can be approximated in C^{1} from inside by smooth sets of the form ∂E_{τ} with H > 0, E′_{τ} = E_{τ}, and

(5.20)  sup_{τ} sup_{∂E_{τ}} |A| < ∞,  ∫_{∂E_{τ}} H^{2} → ∫_{∂E} H^{2} as τ → 0.

Proof. Note that H ≥ 0 on ∂E in the weak sense. We employ the ordinary mean curvature flow ∂x/∂τ = −Hν as developed in [Hu] in order to smooth the boundary while preserving nonnegative curvature. By mollifying, there exists a sequence (Q_{i})_{i≥1} of smooth surfaces with uniformly bounded |A| approximating ∂E from inside in C^{1} and strongly in W^{2,2}, so that

∫_{Q_{i}} H^{2} → ∫_{∂E} H^{2},  ∫_{Q_{i}} H_{−}^{2} → ∫_{∂E} H_{−}^{2}  as i → ∞,

where H_{−} := max(0, −H). Let (Q_{τ}^{i})_{0≤τ<δ_{i}} be the mean curvature flow of Q_{i}. On account of the inequality

\frac{∂|A|^{2}}{∂τ} ≤ Δ|A|^{2} + 2|A|^{4} + C|Rm||A|^{2} + C|∇Rm||A| on Q_{τ}^{i} ,

and higher derivative estimates derived in [Hu], Q_{τ}^{i} exists with uniformly bounded |A| for a uniform time independent of i. Passing smoothly to limits we obtain a flow (Q_{τ})_{0<τ≤δ} with uniformly bounded |A|. We have using the first variation formula and (1.2),

\frac{d}{dτ} ∫_{Q_{τ}^{i}} H^{2} ≤ ∫_{Q_{τ}^{i}} 2H(ΔH + |A|^{2}H + Rc(ν, ν)H) ≤ C ∫_{Q_{τ}^{i}} H^{2}.

Integrating and passing to limits, we obtain ∫_{Q_{τ}} H^{2} ≤ e^{Cτ} ∫_{∂E} H^{2}. Furthermore, because the speed is bounded and using Arzela-Ascoli, we have Q_{τ} → ∂E in C^{1}. So by lower semicontinuity (1.14), we have completed (5.20).

By a variation of this argument, we also have

∫_{Q_{τ}} H_{−}^{2} ≤ e^{Ct} ∫_{∂E} H_{−}^{2} = 0,

so H ≥ 0 on Q_{τ}, and by the strong maximum principle, either H ≡ 0 or H > 0 on Q_{τ}. In the former case, ∂E is a smooth minimal surface. In the latter case, Q_{τ} smoothly foliates an interior neighborhood of ∂E. Using ν_{Q_{τ}} as a calibration as in the proof of Lemma 2.3 and using E′ = E, we see that E′_{τ} = E_{τ}, where E_{τ} is the precompact set bounded by Q_{τ}.

We are now in a position to pass everything to limits to prove the following proposition. M need not be complete.

5.7 H^{2} Growth Formula. Let M be a 3-manifold, E_{0} a precompact open set with C^{1} boundary satisfying

(5.21)  ∫_{∂E_{0}} |A|^{2} < ∞

and (E_{t})_{t>0} a family of open sets solving (†) with initial condition E_{0}. Then for each 0 ≤ r < s,

(5.22)  \begin{aligned}∫_{N_{r}} H^{2} & ≥ ∫_{N_{s}} H^{2} + ∫_{r}^{s} ∫_{N_{t}} (2\frac{|DH|^{2}}{H^{2}} + 2|A|^{2} + 2Rc(ν, ν) − H^{2}) \\ & = ∫_{N_{s}} H^{2} + ∫_{r}^{s} −4πχ(N_{t}) \\ & + ∫_{r}^{s} ∫_{N_{t}} (2\frac{|DH|^{2}}{H^{2}} + \frac{1}{2}(λ_{1} − λ_{2})^{2} + R + \frac{1}{2}H^{2}) . \end{aligned}

provided E_{s} is precompact. In particular, the inner integrals make sense for a.e. t ≥ 0.

The proof consists of passing to limits, using the above lemmas, to get a result for almost every time, then using a restarting argument to get the result for each time.

Proof. 1. In steps 1 and 2, we assume ∂E_{0} is smooth and M possesses a smooth subsolution at infinity, so that the above discussion applies.

Combining together (5.13), Lemma 5.2, (5.18), Fatou’s Lemma, (5.14), Lemma 5.3, and (5.15), we can pass (5.2) to limits as i → ∞. This proves that (5.2) holds as an inequality, with the limiting cylinders Ñ_{t} replacing N_{t}^{ε}, for a.e. 0 < r < s. Since Ñ_{t} is a cylinder for a.e. t, the integral of each geometric quantity Q(x, z) = Q(x) breaks up as

∫Ñ_{t} ϕQ dμÑ_{t} = ∫_{2}^{4} ϕ dz ∫_{N_{t}} Q dμ_{N_{t}} = ∫_{N_{t}} Q dμ_{N_{t}},

and we obtain the first inequality of (5.22) for a.e. 0 < r < s. To see the transition to the second inequality of (5.22), write

2|A|^{2} + 2Rc(ν, ν) − H^{2} = (|A|^{2} − \frac{1}{2}H^{2}) + (R − 2K_{12} + |A|^{2} − H^{2}) + \frac{1}{2}H^{2}

= \frac{1}{2}(λ_{2} − λ_{1})^{2} + R − 2(K_{12} + λ_{1}λ_{2}) + \frac{1}{2}H^{2}

and apply Lemmas 5.4 and 5.5. Thus (5.22) holds for a.e. 0 < r < s.

Recall from (1.10) that N_{t} → N_{s} in C^{1} as t ↗ s. Then the lower semicontinuity of ∫ H^{2} in (1.14) implies (5.22) holds for a.e. r > 0 and all s ≥ r.

2. Next we prove that (5.22) holds at r = 0. Recall from Theorem 1.3(iii) and (1.15) that ∂E′_{0} is C^{1,1} with H ≥ 0 in a weak sense and

∫_{∂E′_{0}} H^{2} ≤ ∫_{∂E_{0}} H^{2}.

So it suffices to prove (5.22) for E′_{0}. There are two cases.

Case 1: If H > 0 somewhere on ∂E′_{0}, then by Lemma 5.6, there is a family of smooth surfaces of the form ∂E_{τ} approximating ∂E_{0} in C^{1}, with H > 0 and E′_{τ} = E_{τ}. By Theorem 3.1, there exists a proper solution (E_{t}^{τ})_{t>0} of (†) with E_{0}^{τ} = E_{τ}. By Lemma 2.4, E_{t}^{τ} evolves smoothly for a short time, so (5.22) is valid for (E_{t}^{τ})_{t≥0} at r = 0.

By (3.1), (5.20), Remark 1 following Theorem 2.1, and the uniqueness given by Theorem 2.2(iii), u^{τ} converges uniformly to u on M \ E′_{0} as τ → 0, with C^{1} convergence of the level sets except at jump times. All quantities in (5.22) converge or are lower semicontinuous to their values for u. This convergence employs the easily proven analogues of (5.13), Lemma 5.2, (5.18), and (5.14) for arbitrary sequences of exact solutions in a 3-manifold; we don’t need a version of (5.15) or Lemma 5.3. In addition, by (5.20),

∫_{∂E_{τ}} H^{2} → ∫_{∂E′_{0}} H^{2},

which proves (5.22) at r = 0 for E′_{0}, and hence for E_{0}, in the case that H > 0 somewhere on ∂E′_{0}.

Case 2: The other possibility is that ∂E′_{0} is a smooth minimal surface. Choose a sequence of smooth functions w_{i} on M such that

w_{i} → 1 in C^{∞}(M),  w_{i} = 1 and \frac{∂w_{i}}{∂ν} > 0 along ∂E_{0},  w_{i} > 1 in M \ E_{0},

and consider the metric g_{i} := w_{i}g. Then H_{i} > 0 along ∂E_{0} and E_{0} is still a strictly minimizing hull with respect to g_{i}. Therefore by Step 2, (5.22) holds at r = 0 for the solution u_{i} of (††) for ∂E_{0} in the metric g_{i}.

As above, passing i → ∞, using (3.1), the Remarks following Theorem 2.1, uniqueness, and the lower semicontinuity of all the quantitites involved, together with the fact that

∫_{∂E_{0}} H_{i}^{2} → ∫_{∂E_{0}} H^{2},

we obtain (5.22) at r = 0 for E′_{0}, and hence for E_{0}, in the case of a minimal surface. Together with Case 1, this proves (5.22) at r = 0 when the initial surface is smooth and there is a subsolution at infinity.

3. Next, let ∂E_{0} be any C^{1} surface satisfying (5.21), but retain the subsolution at infinity. By (5.19), we may approximate ∂E_{0} in C^{1} by smooth surfaces S_{i} lying in M \ E_{0}, such that

(5.23)  ∫_{S_{i}} H^{2} → ∫_{∂E_{0}} H^{2},

as i → ∞. Let (E_{t}^{i})_{t>0} be the proper solution of (†) corresponding to S_{i}. Since S_{i} → ∂E_{0} in C^{1}, there is δ_{i} → 0 such that E_{0} ⊆ E_{0}^{i} ⊆ E_{δ_{i}}, so

E_{t} ⊆ E_{t}^{i} ⊆ E_{t+δ_{i}} for all t ≥ 0,

by the Comparison Theorem 2.2(ii). In particular, u_{i} → u locally uniformly in M \ Ē_{0}, and for each fixed t > 0, ∂E_{t}^{i} is eventually kept uniformly away from S_{i}, so by estimate (3.1) and (2.1), we have

∂E_{t}^{i} → N_{t} in C^{1}, a.e. t > 0.

By Step 2, (5.22) is valid for the approximators (E_{t}^{i})_{t>0} at r = 0, so passing to limits, using (5.23) at r = 0, and lower semicontinuity of all quantities for t > 0, we obtain (5.22) at r = 0 for any initial surface satisfying (5.21) and any M with a smooth subsolution.

4. Finally, the subsolution at infinity can be arranged by employing a conic modification near the edge of the manifold, as in the proof of Theorem 3.1, to prove (5.22) at r = 0 under the stated general hypotheses.

By Lemma 5.5, this result may now be applied at each r ≥ 0 to establish (5.22) for arbitrary 0 ≤ r ≤ s.

The next Theorem now follows from (5.22), recalling from Lemma 1.6 that

|∂E_{t}|^{1/2} = e^{t/2}|∂E_{0}|^{1/2}

provided that E_{0} is a minimizing hull. It is precisely at this point that the exterior region hypothesis declares itself, and it does so through the area rather than through higher order quantities.

5.8 Geroch Monotonicity Formula. Let M be a complete 3-manifold, E_{0} a precompact open set with C^{1} boundary satisfying (5.21), and (E_{t})_{t>0} a solution of (†) with initial condition E_{0}. If E_{0} is a minimizing hull, then

(5.24) m_{H}(N_{s}) ≥ m_{H}(N_{r}) + \frac{1}{(16π)^{3/2}} ∫_{r}^{s} |N_{t}|^{1/2}[16π − 8πχ(N_{t})+

+ ∫_{N_{t}}(2|D log H|^{2} + (λ_{1} − λ_{2})^{2} + R) dμ_{t}] dt

for 0 ≤ r < s, provided E_{s} is precompact.

Remark. According to Lemma 4.1(ii), an exterior region is simply connected. In such a manifold, recall from Lemma 4.2(ii) that a connected surface ∂E_{t} remains connected under the flow, so χ(N_{t}) ≤ 2.^{5} Therefore, we already have the following monotonicity result in the case of a single black hole.

If M̃ is an exterior region satisfying (0.1) and (0.2), having R ≥ 0, and with a connected boundary, then the weak inverse mean curvature flow of the boundary exists forever and has monotone nondecreasing Hawking mass.

5Note that even if N_{0} is a 2-sphere, we cannot exclude the possibility that N_{t} later develops genus.

§6 Multiple Horizons

The purpose of this section is to derive a monotonicity result in the presence of multiple black holes.

6.1 Geroch Monotonicity (Multiple Boundary Components). Let M̃ be an exterior region with R ≥ 0 and satisfying the asymptotic conditions (0.1) and (0.2). For each connected component N of ∂M̃, there exists a flow of compact C^{1,α} surfaces (N_{t})_{t≥0}, such that N_{0} = N, m_{H}(N_{t}) is monotone nondecreasing for all time, and for sufficiently large t, N_{t} satisfies the weak inverse mean curvature flow (†).

The idea is to flow a single boundary component, treating the others as inessential “occlusions” to be slid across, without incurring any loss of Hawking mass. This is achieved by flowing E_{t} so that it nearly touches the rest of the boundary, then jumping to the strictly minimizing hull F of the union of E_{t} with some of the other boundary components. This accomplishes

(6.1)  |∂F | ≥ |∂E_{t_{1}}|,  ∫_{∂F} H^{2} ≤ ∫_{∂E_{t_{1}}} H^{2},

which is sufficient to ensure that the Hawking mass does not decrease during the jump.

Unfortunately, our construction only works on one component at a time. It would be very desirable to have a result involving an additive combination of boundary components. One approach would be to keep track of the mass added during a jump. At the end of §8, we give an example that illustrates the difficulty of doing this.

Proof. Suppose the boundary components of M̃ are N, N_{1}, …, N_{k}. Fill in the boundary by 3-balls

E_{0}, W_{1}, …, W_{k}

to obtain a smooth, complete, boundaryless manifold M. We will construct a nested family of sets (E_{t})_{t≥0} and corresponding locally Lipschitz function u on M such that

(i) E_{t} satisfies the flow except at a finite number of “jump times” 0 < t_{1} < ⋯ t_{p} < ∞,

(ii) Each W_{i} is contained in {u = t_{j}} for some j, and E_{t} swallows at least one W_{i} at each jump time t_{j},

(iii) E_{t} and ∂E_{t} remain connected,

(iv) The Hawking mass m_{H}(∂E_{t}) is monotone nondecreasing.

Write W := ∪_{1≤i≤k}W_{i}. By Lemma 4.1, M is diffeomorphic to R^{3}, and E_{0} ∪ W is a strictly minimizing hull in M.

To start with, let (E_{t})_{t≥0} be the flow of E_{0} by (†). (Later we will modify it past a certain time.) Since M is simply connected and E_{0} is connected, Lemma 4.2 implies that E_{t} and its boundary remain connected.

The case when W is empty is covered by the remarks following Theorem 5.8, so we assume that W is nonempty. Left to its own devices, E_{t} will eventually enter W, and the monotonicity is likely to fail since the scalar curvature there is unknown. So define s_{1} > 0 to be the supremum of the times when E_{t} is disjoint from W. By continuity, we have

E_{s_{1}} ∩ W = ∅, E_{s_{1}}^{+} ∩ W ≠ ∅.

Define t_{1} := s_{1} if Ē_{s_{1}} ∩ W̄ = ∅, otherwise let t_{1} be slightly less that s_{1}.

Since ∂E_{0} is a minimizing hull, ∂E_{t} is connected, and E_{t} stays in the region where R ≥ 0, we have

m_{H}(∂E_{t}) is monotone, 0 ≤ t ≤ t_{1}.

Time to jump.

Now we construct the jump. Let F be the connected component of (E_{t_{1}} ∪ W)′ that contains E_{t_{1}}. Because E_{t_{1}} is so close to W, we see that F contains at least one component of W, and we may write (after relabelling)

F = (E_{t_{1}} ∪ W_{1} ∪ … ∪ W_{j})′,

for some nonempty union of whole components of W.

Since ∂W is a smooth minimal surface, ∂F is disjoint from ∂W by the strong maximum principle. The first inequality of (6.1) is immediate since E_{t_{1}} is a minimizing hull by Lemma 1.4(i).

Let us prove the second inequality of (6.1). By (5.19), choose a sequence of sets E_{i} containing E_{t_{i}} such that ∂E_{i} is smooth and converges to ∂E_{t_{1}} in C^{1}, and

∫_{∂E_{i}} H^{2} → ∫_{∂E_{t_{1}}} H^{2}

We have by (1.15),

∫_{∂E′_{i}} H^{2} ≤ ∫_{∂E_{i}} H^{2}.

Now it can be seen that E′_{i} → F, and ∂E′_{i} → ∂F in C^{1} by Theorem 1.3(ii), so passing to limits and recalling the lower semicontinuity given by (1.14), we obtain (6.1). Then (6.1) implies

m_{H}(∂E_{t_{1}}) ≤ m_{H}(∂F).

In a similar manner, but using Lemma 6.2 below, we will now show that F is a suitable initial condition for restarting the flow. We may approximate ∂E_{t_{1}} in C^{1} by a sequence of smooth surfaces with uniformly bounded mean curvature, of the form ∂U_{i}, where U_{i} contains E_{t_{1}}. Then by (1.15), ∂(U_{i} ∪F)′ is C^{1,1} and itself has uniformly bounded mean curvature, and one easily checks that ∂(U_{i} ∪ F)′ converges to ∂F in C^{1}. By slightly smoothing, we see that ∂F is approximated in C^{1} by smooth surfaces of bounded mean curvature.

Then by Theorem 3.1, (3.1), and Theorem 2.1, there exist sets (F_{t})_{t>0} satisfying (††) with initial condition F_{0} = F. Furthermore, by applying the Gauss-Bonnet Theorem to the approximating sequence, we see

∫_{∂F} |A|^{2} < ∞.

Also, F is a strictly minimizing hull and F_{t} remains connected, so as above, Theorem 5.8 applies to show

m_{H}(F_{t}) is monotone,

at least until F_{t} in turn encounters one of the remaining components of W.

We now replace E_{t} by F_{t} for t > t_{1}. The desired monotonicity holds for a short time past t_{1}, and E_{t} has swallowed at least one component of W. Since W has only a finite number of components, we may continue this procedure inductively to obtain a nested family of sets (E_{t})_{t≥0} satisfying (i)-(iii). This proves the Theorem.

Remark. We have some freedom in choosing the jump time. For example, we could have defined t_{1} to be the first moment when E_{t} together with some component of W possesses a connected minimizing hull, or even the first moment when there is a connected “stationary hull”.

All such schemes implicitly impose grazing (i.e. contact angle zero) boundary conditions for N_{t} along the boundary ∂W. It would be interesting to resolve the ambiguity by deriving the jumping scheme from a suitable minimization principle.

One possibility is to set g = ∞ on W. Like the methods above, this forces u = constant on each F_{i}, and expels the “potential surfaces” N_{t} from the interior of W, much like an electric conductor. However, it is unlikely to yield monotonicity of the Hawking mass since there may be negative R concentrating along the boundary of the obstacle.

Now the promised Lemma, which guarantees that N_{t} always has the necessary smoothness to restart the flow.

6.2 Lemma. Suppose that ∂E is compact and C^{1}, E minimizes

J_{f}(F) := |∂F | + ∫_{F} f

in some open neighborhood U of ∂E, and ess sup_{U} |f| ≤ M. Then ∂E may be approximated in C^{1} on either side by smooth surfaces with |H| ≤ M + ε.

Proof. Choose an open set V with ∂E_{0} ⊂⊂ V ⊂⊂ U. Define g by

g := M + ε in U \ E, g := −M − ε in U ∩ E.

and select a smooth sequence g_{i} with |g_{i}| ≤ M + ε, g_{i} → g in L^{1}(V̄).

Let E_{i} be a minimizer of J_{g_{i}} among all F with F \V̄ = E \V̄. By lower semicontinuity and the compactness theorem for sets of finite perimeter, there exists a set E_{1} and a subsequence with E^{i_{j}} → E_{1}, such that E_{1} minimizes J_{g} subject to the same condition. Since

J_{f}(E) ≤ J_{f}(E_{1}),

the definition of g yields the contradiction

J_{g}(E) < J_{g}(E_{1}),

unless E = E_{1} a.e. Then by the Regularity Theorem 1.3, ∂E_{i} → ∂E in C^{1} locally in V. Excluding extraneous components, we may take the convergence to occur in all of M. The smoothness of g_{i} proves that ∂E_{i} is smooth with |H_{∂E_{i}}| = |g_{i}| ≤ M +ε. To guarantee that E_{i} contains E (say), it suffices to adjoin the condition that g_{i} = −M − ε on U ∩ E.

§7 Asymptotic Regime

By a weak blowdown argument, we show that N_{t} becomes C^{1,α} close to a large coordinate sphere as t → ∞. Then we prove that lim_{t→∞} m_{H}(N_{t}) ≤ m_{ADM}(M) by expanding the Hawking mass as the ADM mass plus lower order terms, using the montonicity formula itself for the analytic control.

Let Ω be the asymptotically flat end of M, embedded in R^{n} as the complement of a compact set K. Let g be the metric of M pulled back to Ω, let δ be the flat metric, and let ∇, ¯∇ be the corresponding connections. Write B_{r}(x) for the balls with respect to g, and D_{r}(x) for standard balls. Let u be a solution of (1.5) in Ω, E^{t} := {u < t} ⊆ Ω.

Fix λ > 0 and define the blown down objects by

Ω^{λ} := λ · Ω, g^{λ}(x) := λ^{2}g(x/λ), u^{λ}(x) := u(x/λ), E_{t}^{λ} := λ · E_{t},

where λ · A := {λx : x ∈ A}. By the scaling property x ↦ λx, t ↦ t, u^{λ} solves (1.5) in Ω^{λ}.

7.1 Blowdown Lemma. Suppose the flat metric on Ω satisfies

(7.1)  |g − δ| = o(1),  | ¯∇g| = o (\frac{1}{|x|}) ,

as |x| → ∞. Let u be a solution of (1.5) on Ω such that {u = t} is compact for all sufficiently large t. Then for some constants c_{λ} → ∞,

u^{λ} − c_{λ} → (n − 1) log |x|,

locally uniformly in R^{n} \{0} as λ → 0, the standard expanding sphere solution in R^{n} \{0}.

As a byproduct, we have the following.

7.2 Proposition. The only solution of (1.5) on R^{n} \ {0} with compact level-sets is the expanding sphere solution given above.

Proof of 7.1 and 7.2. 1. Fix t_{0} so that {u = t} is a compact subset of Ω for all t ≥ t_{0}. The asymptotic condition (7.1) and the comments following Definition 3.3 imply that there is R_{0} > 0 such that

σ(x) ≥ c|x|, dist(x, ∂E_{t_{0}}) ≥ c|x|,

for |x| ≥ R_{0}. By (3.1) with r = min(σ(x), dist(x, ∂E_{t_{0}}), we observe

(7.2)  |∇u(x)| ≤ \frac{C}{|x|},  for all |x| ≥ R_{0}.

Next we want to control the eccentricity of N_{t} for large t. For any surface N, let [r(N), R(N)] be the smallest interval such that N is contained in the annulus D̄_{R} \ D_{r}, and define the eccentricity θ(N) := R(N)/r(N).

Now (7.1) implies that there exists A > 0 and t_{1} such that the family (D_{e^{At}})_{t_{1}≤t<∞} is a subsolution of (⋆), that is, the surfaces are moving faster than inverse mean curvature would dictate. Using these for comparison, we see that

(7.3)  R(N_{t+τ}) ≤ e^{Aτ} R(N_{t}),  t ≥ t_{1}, τ ≥ 0.

Next suppose r = r(N_{t}) ≥ R_{0}. Now u = t somewhere on ∂D_{r}, and so by (7.2), there is C_{2} such that u > t − C_{2} everywhere on ∂D_{r}. Therefore N_{t−C_{2}} does not meet ∂D_{r}. By 4.2(i), N_{t−C_{2}} cannot have any components outside of D_{r}, so R(N_{t−C_{2}}) ≤ r. Combining with (7.3),

(7.4)  R(N_{t}) ≤ e^{AC_{2}}R(N_{t−C_{2}}) ≤ e^{AC_{2}}r(N_{t}),  t ≥ t_{2},

for some t_{2}.

2. Now let λ_{i} be any sequence converging to zero. The estimates (7.2), (7.3), and (7.4) are scale-invariant, so they are valid for N_{t}^{λ_{i}}, on the complement of a subset shrinking to {0} as i → ∞.

By (7.2), Compactness Theorem 2.1, and the Remarks following it, there exists a subsequence (λ_{i_{j}}), numbers c_{j} → ∞, and a solution v of (1.5) in R^{n} \ {0}, such that

u^{λ_{i_{j}}} − c_{j} → v locally uniformly in R^{n} \ {0},

with local C^{1} convergence of the level sets. Define P_{t} := ∂{v < t}. By the eccentricity estimate (7.4), each nonempty P_{t} is a compact subset of R^{n} \ {0} with

θ(P_{t}) ≤ e^{AC_{2}}.

Now (7.3) implies in the limit that v is not constant, so some level-set is nonempty, say P_{t_{3}}. Then (7.3), together with the fact that N_{t} is nonempty and compact for all sufficiently large t, implies that P_{t} is nonempty and compact for −∞ < t < ∞.

Thus, v has compact level sets. According to Proposition 7.2, then, P_{t} is a family of expanding, round spheres and u has the form exhibited. Since this is true for a subsequence of any subsequence, it follows that the full sequence converges, proving Lemma 7.1.

3. It remains to prove Proposition 7.2. Let v be a solution of (1.5) in R^{n} \ {0} with compact level sets P_{t}. Using expanding spheres as barriers, we see that

θ(P_{t}) is nonincreasing.

Now, using the estimates above, we can blow up a subsequence (P_{t}^{λ_{i}})_{−∞<t<∞}, as λ_{i} → ∞, to obtain a solution (Q_{t})_{−∞<t<∞} with nonempty, compact level sets and

θ(Q_{t}) ≡ θ_{0}, −∞ < t < ∞,

where θ_{0} := sup_{t} θ(P_{t}). If θ_{0} > 1, then Q_{0} lies between ∂D_{r} and ∂D_{θ_{0}r} for some r, and is tangent to each of them, without being equal to either one. By perturbing ∂D_{r} outward and ∂D_{θ_{0}r} inward, and applying the strong maximum principle to the smooth flows, and the weak maximum principle, Theorem 2.2(ii), to the nonsmooth flow Q_{t}, we find that θ(Q_{t}) decreases, a contradiction. This proves that θ_{0} = 1, which shows that P_{t} is a round sphere for all t, as claimed in the Proposition.

The ADM mass is defined in the following lemma. Here U denotes a precompact open set with smooth boundary, ν is the outward unit normal of U with respect to g, and dμ the surface measure of ∂U with respect to g.

7.3 ADM Lemma. Suppose R ≥ 0 on M, and the asymptotic region Ω is embedded as the complement of a compact set in R^{3}, with flat metric δ.

(i) (see [ADM]) If δ satisfies

cδ ≤ g ≤ Cδ in Ω, ∫_{Ω} | ¯∇g|^{2} < ∞,

then the limit

m_{ADM}(g, δ) := lim_{U→M} \frac{1}{16π} ∫_{∂U} g^{ij}( ¯∇_{j}g_{ik} − ¯∇_{k}g_{ij})ν^{k} dμ

exists, and is finite if and only if ∫_{M} R < ∞. Here χ_{U} → χ_{M} locally uniformly. (ii) (Bartnik [Ba2], Chruściel [Ch3]) If δ satisfies

|g − δ| ≤ C|x|^{−1/2−α},  | ¯∇g| ≤ C|x|^{−3/2−α},  x ∈ Ω,

for some α > 0, then m_{ADM} is a geometric invariant of g, independent of the choice of δ.

Proof. (i) We observe that the vector field

Y^{k} := g^{kl}g^{ij}( ¯∇_{i}g_{jl} − ¯∇_{l}g_{ij}) = g^{ij}Γ_{ij}^{k} − Γ_{jl}^{j}g^{lk}

satisfies

div Y = R ± C|Γ|^{2},

where the divergence is taken with respect to g. Integrating by parts yields the result.

(ii) See [Ba2,Ch3].

In order to effect our mass comparison at infinity, we recall the conditions from the introduction, namely

(0.1)-(0.2)  |g − δ| ≤ \frac{C}{|x|},  | ¯∇g| ≤ \frac{C}{|x|^{2}} ,  Rc ≥ −\frac{Cg}{|x|^{2}} .

Note that this implies directly (without integration by parts as needed with weaker conditions) that the ADM flux integral is finite.

7.4 Asymptotic Comparison Lemma. Assume that the asymptotic region of M satisfies (0.1), and let (E_{t})_{t≥t_{0}} be a family of precompact sets weakly solving (⋆) in M. Then

lim_{t→∞} m_{H}(N_{t}) ≤ m_{ADM}(M).

The proof consists of a straightforward linearization of ∫ H^{2} in terms of the perturbation from flatness, together with the integration by parts formula (7.15), which is possibly related to the “curious cancellation” in [Ba1,Ch]. We use (7.2) and (5.22) to show that m_{H} is bounded, which gives us some analytic control of the sequence via Geroch Monotonicity, which allows us to refine the estimates. We ultimately obtain convergence of |A|^{2} in L^{2}.

Proof. 1. Define r = r(t) by |N_{t}| = 4πr^{2}. Then |N_{t}^{1/r}|_{g^{1/r}} = 4π, so Lemma 7.1 implies that

(7.5)  N_{t}^{1/r(t)} → ∂D_{1} in C^{1} as t → ∞.

Let h be the restriction of g to the moving surface and let ε be the restriction of the flat metric δ to it. Let ν be the exterior unit normal, ω the unit dual normal, A the second fundamental form, H the mean curvature, all with respect to g. Define ν̄, ω̄, Ā, H̄ correspondingly, with respect to δ.

We begin with a series of approximations. Write p_{ij} = g_{ij} − δ_{ij}. Let us restrict our attention to t sufficiently large that |p| ≤ 1/10 on N_{t}. Then we have

(7.6)  h^{ij} − ε^{ij} = −h^{ik}p_{kl}h^{lj} ± C|p|^{2},  g^{ij} − δ^{ij} = −g^{ik}p_{kl}g^{lj} ± C|p|^{2},

for the inverse matrices. The bars indicate the norm with respect either to g or to δ – it does not matter which. Note that

ω = \frac{ω̄}{|ω̄|_{g}} , ν^{i} = g^{ij}ω_{j}.

Then we have

ω̄_{i} = ω_{i} ± C|p|,  ν̄^{i} = ν^{i} ± C|p|,  1 − |ω̄|_{g} = \frac{1}{2}ν^{i}ν^{j}p_{ij} ± C|p|^{2}, (7.7)

dμ − dμ̄ = (\frac{1}{2}h^{ij}p_{ij} ± C|p|^{2}) dμ, (7.8)

Γ_{ij}^{k} = \frac{1}{2}g^{kl}(∇_{i}p_{jl} + ∇_{j}p_{il} − ∇_{l}p_{ij}) ± C|p||∇p|, ∇p = ¯∇p ± C|p||∇p|, (7.9)

where Γ_{ij}^{k} are the Christoffel symbols ∇ − ¯∇. We have the formula

(7.10)  |ω̄|_{g}A_{ij} = Ā_{ij} − ω̄_{k}Γ_{ij}^{k}.

Compute

H − H̄ = h^{ij}A_{ij} − ε^{ij}Ā_{ij} = (h^{ij} − ε^{ij})A_{ij} + ε^{ij}A_{ij}(1 − |ω̄|_{g}) + ε^{ij}(|ω̄|_{g}A_{ij} − Ā_{ij}).

For the second term we have, using (7.7) and (7.6),

ε^{ij}A_{ij}(1 − |ω̄|_{g}) = \frac{1}{2}Hν^{i}ν^{j}p_{ij} ± C|p|^{2}|A|,

and for the third, using (7.10), (7.6), (7.7), and (7.9),

ε^{ij}(|ω̄|_{g}A_{ij} − Ā_{ij}) = −ε^{ij}ω̄_{k}Γ_{ij}^{k}

= −\frac{1}{2}h^{ij}ω_{k}g^{kl}(∇_{i}p_{jl} + ∇_{j}p_{il} − ∇_{l}p_{ij}) ± C|p||∇p|

= −h^{ij}ν^{l}∇_{i}p_{jl} + \frac{1}{2}h^{ij}ν^{l}∇_{l}p_{ij} ± C|p||∇p|.

Plugging these in and using (7.6), we get

H − H̄ = −h^{ik}p_{kl}h^{lj}A_{ij} + \frac{1}{2}Hν^{i}ν^{j}p_{ij} − h^{ij}ν^{l}∇_{i}p_{jl} + \frac{1}{2}h^{ij}ν^{l}∇_{l}p_{ij}

± C|p||∇p| ± C|p|^{2}|A|,

and so

|H − H̄| ≤ C|p||A| + C|∇p|, |H^{2} − H̄^{2}| ≤ C|p||A|^{2} + C|∇p|^{2} + C|∇p||A|,

and using this together with (7.8),

H̄^{2}(dμ − dμ̄) = (\frac{1}{2}H^{2}h^{ij}p_{ij} ± C|p|^{2}|A|^{2} ± C|∇p|^{2}) dμ.

We are now in a position to estimate ∫ H^{2}. Our first task is to isolate the leading term 16π from the correction of order 1/r which gives the mass. We have from [Wi] that in R^{3},

∫_{N} H̄^{2} dμ̄ ≥ 16π.

From this inequality and the previous three, we find for sufficiently large t,

(7.11) \begin{aligned}∫_{N_{t}} H^{2} dμ & = ∫_{N_{t}} H̄^{2}dμ̄ + H̄^{2}(dμ − dμ̄) + 2H(H − H̄) − (H − H̄)^{2} dμ \\ & ≥ 16π + ∫_{N_{t}} \frac{1}{2}H^{2}h^{ij}p_{ij} − 2Hh^{ik}p_{kl}h^{lj}A_{ij} + H^{2}ν^{i}ν^{j}p_{ij} \\ & − 2Hh^{ij}ν^{l}∇_{i}p_{jl} + Hh^{ij}ν^{l}∇_{l}p_{ij} − C|p|^{2}|A|^{2} − C|∇p|^{2}. \end{aligned}

2. Next we show that m_{H}(N_{t}) remains bounded. As remarked in the proof of Lemma 7.1, (0.1) implies that

(7.12)  |H| = |∇u| ≤ \frac{C}{|x|} ≤ \frac{C}{r}  on N_{t},

for a.e. sufficiently large t, where we have recalled (1.12), and used (7.5) to relate |x| to r. Using this with (5.22),

∫_{s}^{s+1} ∫_{N_{t}} |A|^{2} = ∫_{s}^{s+1} ∫_{N_{t}} \frac{1}{2}(λ_{1} − λ_{2})^{2} + \frac{1}{2}H^{2}
≤ ∫_{N_{s}} H^{2} + ∫_{s}^{s+1} 4πχ(N_{t})
≤ 4πC + 8π,

so we may select a subsequence t_{i} → ∞ such that

sup_{i} ∫_{N_{t_{i}}} |A|^{2} < ∞.

Then we estimate from (7.11), (0.1), and (7.12),

∫_{N_{t_{i}}} H^{2} ≥ 16π − C ∫_{N_{t_{i}}} |p||A|^{2} + |∇p||H| + |∇p|^{2} ≥ 16π − \frac{C}{r}

so by the definition and monotonicity of m_{H}(N_{t}),

sup_{t} m_{H}(N_{t}) < ∞.

3. In particular, this implies by the monotonicity formula (5.24),

∫_{t_{0}}^{∞} e^{t/2} ∫_{N_{t}}(λ_{1} − λ_{2})^{2} + \frac{|DH|^{2}}{H^{2}} < ∞,

so we can pick a new subsequence t_{ℓ} → ∞ such that, writing N_{ℓ} := N_{t_{ℓ}},

∫_{N_{ℓ}}(λ_{1} − λ_{2})^{2} + \frac{|DH|^{2}}{H^{2}} → 0,

and (by scaling) a corresponding result for N_{ℓ}^{1/r_{ℓ}}. Since H ≤ C/r, we can use Rellich’s theorem as in the proof of 5.12 to show that

H_{N_{ℓ}^{1/r_{ℓ}}} → H_{∂D_{1}} = 2 in L^{2}(∂D_{1}),

where N_{ℓ}^{1/r_{ℓ}} is written as a graph over ∂D_{1} for large i. Rescaling, this implies

(7.13)  H_{N_{ℓ}} = \frac{2}{r} + f_{ℓ}  on N_{ℓ},

where ∫_{N_{ℓ}} f_{ℓ}^{2} → 0. Also,

∫_{N_{ℓ}} |A − \frac{H}{2} h|^{2} = \frac{1}{2} ∫_{N_{ℓ}}(λ_{1} − λ_{2})^{2} → 0,

so

(7.14)  A_{N_{ℓ}} = \frac{h}{r} + g_{ℓ}  on N_{ℓ},

where ∫_{N_{ℓ}} g_{ℓ}^{2} → 0. In particular,

sup_{ℓ} ∫_{N_{ℓ}} |A|^{2} < ∞.

4. Thus we may estimate from (7.11), (7.13) and (7.14),

32πm_{H}(N_{ℓ}) = r_{ℓ}(16π − ∫_{N_{ℓ}} H^{2})

≤ \frac{C}{r_{ℓ}} + r_{ℓ} ∫_{N_{ℓ}} −\frac{1}{2}H^{2}h^{ij}p_{ij} + 2Hh^{ik}p_{kl}h^{lj}A_{ij} − H^{2}ν^{i}ν^{j}p_{ij}

+ 2Hh^{ij}ν^{l}∇_{i}p_{jl} − Hh^{ij}ν^{l}∇_{l}p_{ij},

≤ \frac{C}{r_{ℓ}} + η_{ℓ} + ∫_{N_{ℓ}} −\frac{2}{r_{ℓ}} h^{ij}p_{ij} + \frac{4}{r_{ℓ}} h^{ij}p_{ij} − \frac{4}{r_{ℓ}} ν^{i}ν^{j}p_{ij}

+ 4h^{ij}ν^{l}∇_{i}p_{jl} − 2h^{ij}ν^{l}∇_{l}p_{ij},

where η_{ℓ} is an error term of the form

η_{ℓ} = C ∫_{N_{ℓ}} |f_{ℓ}||p| + |g_{ℓ}||p| + r_{ℓ}|f_{ℓ}||∇p|.

By (0.1), we see that η_{ℓ} → 0. We observe by the integration by parts formula (5.17), that

(7.15) ∫_{N_{ℓ}} h^{ij}ν^{l}∇_{i}p_{jl} = ∫_{N_{ℓ}} Hν^{j}ν^{l}p_{jl} − h^{ij}p_{jk}h^{kl}A_{li} = ∫_{N_{ℓ}} \frac{2}{r_{ℓ}} ν^{j}ν^{l}p_{jl} − h^{ij}p_{ij} + η′_{ℓ},

where again, η′_{ℓ} → 0 by (0.1). Applying this to the previous equation we get

32πm_{H}(N_{ℓ}) ≤ \frac{C}{r_{ℓ}} + η_{ℓ} + 2η′_{ℓ} + ∫_{N_{ℓ}} 2h^{ij}ν^{l}∇_{i}p_{jl} − 2h^{ij}ν^{l}∇_{l}p_{ij}.

Using (7.9), this differs from the definition of the ADM mass in Lemma 7.3 by at most

η′′_{ℓ} = C ∫_{N_{ℓ}} |p||∇p|,

which again converges to zero by (0.1). Applying the monotonicity formula (5.24), this yields

sup_{t≥0} m_{H}(N_{t}) ≤ m_{ADM}(M).

§8 Proof of Main Theorem

In this section we prove the rigidity claim, and assemble the pieces to prove the Main Theorem. At the end of the section, we give an example relating to multiple black holes and the nonlocality of mass.

The following lemma is useful for the no-boundary case.

8.1 Lemma. Let M be a complete, asymptotically flat 3-manifold with no boundary. For any x ∈ M there is a locally Lipschitz solution u of (1.5) on M \ {x} with u → −∞ near x and u → ∞ at infinity. Furthermore, m_{H}(N_{t}) ≥ 0 for all t.

Proof. By Theorem 3.1 and Remark 1 following it, for each ε > 0, there exists u^{ε} solving (††) with the initial condition B_{ε}(x). We may use the eccentricity estimate and other techniques of Lemma 7.1, to show that there exists a subsequence ε_{i} → 0, a sequence c_{i} → ∞, and a function u defined on M \ {x} such that u^{ε_{i}} − c_{i} → u locally uniformly,

|∇u(y)| ≤ \frac{C}{dist(y, x)} in B_{1}(x),

and N_{t} is nonempty and compact for all t, with N_{t} nearly equal to ∂B_{e^{t/(n−1)}} as t → −∞.

Recalling Theorem 5.8, we have

m_{H}(N_{t+c_{i}}^{ε_{i}}) ≥ m_{H}(∂B_{ε_{i}}(x)), −c_{i} ≤ t < ∞.

Passing N_{t+c_{i}}^{ε_{i}} → N_{t} for a.e. t, and recalling the upper semi-continuity of Hawking mass, we obtain m_{H}(N_{t}) ≥ 0, as required.

Proof of Main Theorem. 1. Let M be an exterior region satisfying (0.1). If M has no boundary, then by Lemma 7.4, the solution given in Lemma 8.1 proves m_{ADM}(M) ≥ 0.

If M has a boundary, let N be any boundary sphere. By Theorem 6.1, there exists a flow (N_{t})_{t≥0} such that N_{0} = N and m_{H}(N_{t}) is monotone. Together with Lemma 7.4 this proves that

m_{ADM}(M) ≥ \sqrt{\frac{|N|}{16π}}

which is (0.4).

2. To complete the proof, we must consider the case of equality. We treat the case when ∂M is nonempty; the treatment of the no-boundary case is essentially identical. Then m_{H}(N_{t}) equals a constant, m, for all t ≥ 0. By the construction in Theorem 6.1, N_{t} solves the flow except for a finite number of times, remains connected, and stays in the regions where R ≥ 0. So all the growth terms in (5.24) are nonnegative, and therefore zero, for a.e. t. In particular, recalling from Lemma 5.1 that H > 0 a.e. on N_{t} for a.e. t, we have ∫_{N_{t}} |DH|^{2} = 0 for a.e. t, and therefore by (1.10) and lower semicontinuity,

∫_{N_{t}} |DH|^{2} = 0 for all t.

Therefore

H_{N_{t}}(x) = H(t),  a.e. x ∈ N_{t},  for all t ≥ 0,

that is, N_{t} has constant mean curvature. Since H is locally bounded on M \ E_{0} and N_{t} has locally uniform C^{1} estimates, it follows by elliptic theory that N_{t} is smooth for each t, with estimates that are locally uniform for t ≥ 0. Similar considerations apply to N_{t}^{+}, t ≥ 0.

If there is a jump at time t, either natural or constructed, then N_{t}^{+} ≠ N_{t}, and H = 0 on a portion of N_{t}^{+}, so N_{t}^{+} is a minimal surface, disjoint from ∂M by the strong maximum principle. This contradicts the assumption that M contains no compact minimal surfaces in its interior. Therefore N_{t} = N_{t}^{+} for all t ≥ 0 (so in particular there is at most one boundary component!).

This shows H > 0 for t > 0, and a convergence argument shows that H(t) is locally uniformly positive for t > 0.

By Lemma 2.4, for each t > 0 there is some maximal T > t such that (N_{s})_{t≤s<T} is a smooth evolution. By the regularity derived above, N_{s} has uniform space and time derivatives as s ↗ T, so the evolution can be continued smoothly past t = T. This shows T = ∞ and the entire flow is smooth.

By the vanishing of the growth terms in (5.24), we have

λ_{1} = λ_{2} = \frac{H}{2} , R = 0,

on each N_{t}. Then (1.3) says

\frac{d}{dt}H^{2} = −2|A|^{2} − 2Rc(ν, ν),

showing Rc(ν, ν) is constant on each N_{t}, and then

K = K_{12} + λ_{1}λ_{2} = −Rc(ν, ν) + λ_{1}λ_{2},

showing that K is constant on N_{t}, and N_{t} is isometric to a round sphere.

Since the normal speed, 1/H, is constant on each N_{t}, the metric on M takes the warped product form

g = \frac{1}{H^{2}} dt^{2} + h_{N_{t}}

where t = u(x). We define r = r(t) by imposing

h_{N_{t}} = r^{2} dS

where dS is the standard metric on S^{2}. Then differentiating the relation

e^{t}|N_{0}| = |N_{t}| = 4πr^{2}

yields

dt = 2\frac{dr}{r} .

Now

m = \frac{|N_{t}|^{1/2}}{(16π)^{3/2}} (16π − ∫_{N_{t}} H^{2})

from which we obtain

H^{2} = \frac{4}{r^{2}} − \frac{8m}{r^{3}}.

Substituting these relation above, we get

g = \frac{dr^{2}}{1 − 2m/r} + r^{2}dS

which is isometric to R^{3} or (taking its exterior region) to one-half of the Schwarzschild manifold (see [HE, p.149]).

Nonlocal Mass Example. We are now in a position to give an example that sheds light on the nonlocality of mass distribution. In the artificial-jump construction of Theorem 6.1, it is tempting to include a contribution from the other boundary components as they are jumped over. However, the following example shows that not all of the extra mass is acquired at the moment of jumping.

Let M be half of a Schwarzschild manifold of mass m > 0 and boundary ∂F_{1}, and select a point p in M. Consider p as the site of an infinitely small black hole of mass zero. Let (N_{t})_{t>0} be the flow starting from p and jumping over ∂F_{1} at t = t_{1} according to the procedure of Theorem 6.1. We have

0 = m_{H}(N_{0}) < m_{H}(N_{t_{1}}) < m_{H}(N_{t_{1}}^{+}) < m_{H}(N_{∞}) = m.

The middle inequality represents the jump. The first and third inequalities are strict because of the rigidity statement proven above, for otherwise M would be rotationally symmetric around p, which is impossible.

So, from the point of view of p, part of the mass is located in the “field” fore and aft of the black hole ∂F_{1}, whereas from the point of view of ∂F_{1}, all the mass is on the horizon.

This example may easily be modified so that the horizon at p has positive size (by summing two dissimilar Laplace kernels), and similar considerations apply. We would like to thank Professors Friedman, Geroch, and Wald for conversations about this example.

§9 Applications to Quasi-Local Mass

In many circumstances, the double integral on the right-hand side of the Geroch Monotonicity Formula (5.24) can be estimated concretely in terms of local information to yield an explicit lower bound for the ADM mass. In this section, we examine the implications of this idea for the quasi-local mass, or gravitational capacity, put forward by Bartnik. In particular, we show that the Bartnik capacity satisfies the positivity and exhaustion properties conjectured in [Ba3,Ba4,Ba5]. We end with some speculations about how the ADM mass might control the convergence of ADM-mass minimizing sequences of 3-manifolds.

Let us first give an analogy from electrostatics [Ba4, p.123-4]. If E is a bounded set in R^{n}, n ≥ 3, consider the class of potentials such that

(9.1)  −Δu ≥ 0 in R^{n},  u ≥ 1 in E,  u → 0 as |x| → ∞,

where ρ ≡ −Δu is the charge density. The total charge Q is given by

Q(u) = ∫ ρ = lim_{r→∞} ∫_{∂B_{r}} −u_{ν},

analogous to the ADM mass. One definition of the capacity of E is the smallest total charge consistent with (9.1). It is attained by the solution of the exterior Dirichlet problem with u = 1 on ∂E.

Let (Ω, g) be a Riemannian 3-manifold; we assume that Ω has no boundary and its metric closure is compact. We define an admissible extension of Ω to be an isometry between Ω and a bounded open set in an exterior region M as defined in the Main Theorem, to wit: a complete, connected, asymptotically flat 3-manifold with one end, satisfying (0.1)-(0.2), having R ≥ 0, and possessing a minimal, compact boundary (possibly empty) but no other compact minimal surface. If Ω possesses an admissible extension, we call Ω admissible.

In analogy with the electrostatic capacity, we define the Bartnik gravitational capacity of Ω by

c_{B}(Ω) := inf{m_{ADM}(M) : M is an admissible extension of Ω}.

Note that Bartnik’s original definition required M to be free of minimal surfaces (and hence diffeomorphic to R^{3}), whereas our definition allows minimal surfaces on the boundary.

Thus Ω may surround one or more horizons, which contribute to the mass through their effect on the shape of Ω. But we require Ω to lie outside of these horizons; otherwise it would be unclear what part of Ω should be exposed and what part shielded. In addition, our definition permits extra horizons in the infinite component of M \ Ω. (See figure.) Though that the ADM mass depends only on the infinite component, the finite components affect the admissibility of M. Our definition is a-priori less than or equal to Bartnik’s, and conjecturally equal.

Admissible extension of Ω.

Note that c_{B} trivially has the monotonicity property [Ba3]

Ω̃ ⊆ Ω implies c_{B}(Ω̃) ≤ c_{B}(Ω).

Bartnik conjectured in [Ba3,Ba4,Ba5] that c_{B} > 0 except on subsets of Euclidean space. Now if Ω happens to be a minimizing hull in the extension M, then by the proof of Theorem 6.1 and Lemma 7.4, we have

m_{ADM}(M) ≥ m_{H}(∂Ω).

If we only had to deal with extensions of this kind, then c_{B} would immediately acquire a positive lower bound, namely m_{H}. By taking some care about the minimizing hull property, we are able to prove the following.

9.2 Positivity Property. Let Ω be admissible. Then c_{B}(Ω) > 0 unless Ω is locally isometric to Euclidean R^{3}.

Careful examination of the following proof should give a small, but explicit lower bound for c_{B}(Ω) in terms of the scalar curvature in a small region (compare [Ba5, eqn. (50)]). It remains open whether Ω can be isometrically embedded in R^{3}. (See the conjecture at the end of the section.)

Proof. Let x be a point in Ω where the metric is not flat. Let M_{0} be an admissible extension of Ω, and let (E_{t})_{t∈R} be the solution of inverse mean curvature flow in M_{0} \ {x} given by Lemma 8.1. Since Ω is not flat near x, the simple connectedness of B_{r}(x), (5.24), and the argument used to prove the rigidity in §8 show that

m_{H}(∂E_{t}) > 0

for any t ∈ R. It will suffice to show that the initial part of the flow is valid up to a time t_{0} that is independent of the extension.

Choose r > 0 such that B_{r}(x) is a topological ball and B_{3r}(x) ⊆ Ω. By the monotonicity formula for minimal surfaces [S, p.85], any properly embedded minimal surface in B_{3r}(x) \ B̄_{r}(x) that meets ∂B_{2r}(x) has a certain minimum area c. Select t_{0} so that E_{t_{0}} ⊆ B_{r}(x) and |∂E_{t_{0}}| < c. By Lemma 1.4(i), E_{t_{0}} is a minimizing hull in Ω. We will prove that that this remains true in any extension.

Let M be any other admissible extension of Ω, and let P be a surface that minimizes area among all surfaces in M that separate E_{t_{0}} from the asymptotically flat end of M. Then P \ Ē_{t_{0}} is a properly embedded minimal surface in M \ Ē_{t_{0}}, and |P | ≤ |∂E_{t_{0}}| < c, so P cannot meet ∂B_{2r}(x). Since M contains no closed minimal surfaces besides ∂M, P must lie in B_{2r}(x) ⊆ Ω. Since E_{t_{0}} is a minimizing hull in Ω, we have

|∂E_{t_{0}}| ≤ |P |,

which shows that ∂E_{t_{0}} is a minimizing hull in M.

Therefore, by Lemma 7.4 and the argument of Theorem 6.1, there is a flow starting at E_{t_{0}} that proves

m_{H}(∂E_{t_{0}}) ≤ m_{ADM}(M).

Since M was an arbitrary extension of Ω, this shows

0 < m_{H}(∂E_{t_{0}}) ≤ c_{B}(Ω).

9.3 Exhaustion Property. Let M be asymptotically flat satisfying (0.1). If Ω_{i} is a sequence of bounded sets such that χ_{Ω_{i}} → χ_{M} locally uniformly, then c_{B}(Ω_{i}) → m_{ADM}(M).

Proof. Select R_{i} → ∞ such that D_{R_{i}} ⊆ Ω_{i}, where D_{R} denotes the coordinate ball. By (0.1), D_{R_{i}} \ D_{R_{i}/25} is C^{1} close to flat for large i. It follows by an argument similar to the above that D_{R_{i}/25} is a minimizing hull (in the sense employed above) in any extension M̃ of D_{R_{i}}. By the argument of Theorem 6.1, there exists a flow in M̃ that proves

m_{H}(D_{R_{i}/25}) ≤ m_{ADM}(M̃).

Taking the infimum over M̃, this proves

m_{H}(D_{R_{i}/25}) ≤ c_{B}(Ω_{i}),

which does not exceed m_{ADM}(M). A direct calculation similar to Lemma 7.4 (but easier) proves m_{H}(D_{R_{i}/25}) → m_{ADM}(M), yielding the Proposition.

Mass Minimization. Let Ω be a fragment of a Riemannian 3-manifold as above. Does there exist an admissible extension M of Ω that achieves the infimum c_{B}(M), and what Euler-Lagrange equation does it solve? Our discussion is inspired by conversations with R. Bartnik, H. Bray, M. Gromov, and R. Ye.

Bartnik has made the following conjecture, which we have adapted slightly to our setting of manifolds with minimal boundary. Assume that Ω is connected with smooth boundary, possibly consisting of several components. Let U denote the component of M \Ω̄ that contains infinity. According to our definition, ∂U is a disjoint union of ∂U ∩∂Ω together with extra boundary horizons ∂U ∩ ∂M.

Static Minimization Conjecture (Bartnik [Ba4, p.126]. The infimum c_{B}(Ω) is realized by an admissible extension M of Ω such that U is matter-free, static, and contains no boundary horizons.^{6} The metric is smooth on U and C^{0,1} across ∂U, with nonnegative distributional scalar curvature on ∂U.

Let us explain this heuristically; details can be found in [Ba3,Ba4,Ba5]. Consider (U, g) as time-symmetric initial data for the Einstein equation, and let L be a space-time development. Matter-free asserts that the 4-dimensional Ricci curvature of L vanishes; static asserts that L possesses a timelike Killing field orthogonal to a foliation by hypersurfaces. The idea is that once the extra mass-energy has been squeezed out, there is nothing left to support matter fields or gravitational dynamics. The fact that U has vanishing second fundamental form allows one to deduce that L is static and that U is one of the hypersurfaces in the foliation (see [Ba5]). Fortified by the Penrose Inequality, we expect that horizons in ∂U would contribute extra ADM mass and therefore do not arise; thus ∂U equals ∂U ∩ ∂Ω.

In Riemannian terms, the Euler-Lagrange equation then becomes [Ba4, p.115], [Ba5, (67)-(68)],

(9.2)  \{ \begin{matrix}R = 0,  Rc = \frac{D^{2}ϕ}{ϕ} & in U, \\ H^{+} = H^{−},  g^{+} = g^{−} & along ∂U. \end{matrix}

for some function ϕ on U. Here H^{+}, H^{−}, g^{+}, g^{−} denote the mean curvature and the metric of ∂U as induced from Ω and from U, respectively.

In the electrostatic analogy, a minimizing potential u of the Dirichlet integral on R^{3}\K is harmonic: a static, sourceless solution of the wave equation. Note that u, like g, is not C^{1} across the boundary.

Let us now consider how minimizing sequences for c_{B}(Ω) might converge. To treat the simplest case, suppose first that M_{i} is a sequence of asymptotically flat 3-manifolds with nonnegative scalar curvature and

m_{ADM}(M_{i}) → 0.

Based on the equality case of the Positive Mass Theorem, we expect that M_{i} converges to flat R^{3} in some sense.

6We have added this last proviso.

But how exactly? In a near-Euclidean Sobolev-space setting, Bartnik shows that the mass controls the metric in a W^{1,2} sense [Ba5, Thm.4]. On the other hand, more globally, we must certainly exclude the region behind the horizons, whose total area is thankfully at most 16πm^{2} according to the Penrose inequality in the definitive form given by Bray [B2]. One might then expect that the resulting exterior regions converge in the Gromov-Hausdorff topology, but this is not quite so.

For example, even with no horizons, M_{i} can contain arbitrarily many long, thin cylindrical spikes that look like S^{2}(ε)×[0, L] capped off by a hemisphere S_{+}^{3}(ε), without violating positive scalar curvature or contributing significantly to the ADM mass. Neither the number nor the volume of these spikes can be bounded. But inspired by the Penrose inequality, we speculate that the spikes are contained in a set whose total perimeter can be controlled by some multiple of the ADM mass. This leads to the following conjecture.

Conjecture. Suppose M_{i} is a sequence of asymptotically flat 3-manifolds with ADM mass tending to zero. Then there is a set Z_{i} ⊆ M_{i} such that |∂Z_{i}| → 0 and M_{i} \ Z_{i} converges to R^{3} in the Gromov-Hausdorff topology.

The process of resolving this conjecture is likely to produce analytic tools that can be applied to the more general question of the convergence of minimizing sequences in the case c_{B}(Ω) > 0, and lead to a solution of Bartnik’s Static Minimization Conjecture.

References

[A] F. Almgren, Existence and Regularity Almost Everywhere of Solutions to Elliptic Vari-

ational Problems with Constraints, Mem. Amer. Math. Soc. #165, Providence, 1976.

[ADM] R. Arnowitt, S. Deser and C. Misner, “Coordinate Invariance and Energy Expressions in General Relativity,” Phys. Rev. 122 (1961) 997–1006.

[AM] A. Ashtekar and A. Magnon, “From i_{0} to the 3 + 1 Description of Spatial Infinity,” J. Math. Phys. 25 (1984) 2682–2690.

[Ba1] R. Bartnik, “Existence of Maximal Surfaces in Asymptotically Flat Spacetimes,” Comm. Math. Phys. 94 (1984) 155–175.

[Ba2] R. Bartnik, “The Mass of an Asymptotically Flat Manifold,” Comm. Pure Appl. Math. 39 (1986) 661–693.

[Ba3] R. Bartnik, “New Definition of Quasi-Local Mass,” Phys. Rev. Lett. 62 (1989) 2346.

[Ba4] R. Bartnik, “Results and Conjectures in Mathematical Relativity,” Miniconf. on Geom. and Phys. (Canberra, 1989), Proc. Centre Math. Anal., Austr. Nat. Univ., Canberra, 1989, 110–140.

[Ba5] R. Bartnik, “Energy in General Relativity,” Tsing Hua lectures on Geometry and Analysis (Hsinchu, 1990–1991), Internat. Press, Cambridge, MA, 1997, 5–27.

[Ba6] R. Bartnik, “Quasi-Spherical Metrics and Prescribed Scalar Curvature,” J. Diff. Geom. 37 (1995) 31–71.

[BT] R. C. Bassanezi and I. Tamanini, “Subsolutions to the Least Area Problem and the ‘Minimal Hull’ of a Bounded Set in R^{n},” Ann. Univ. Ferrara Sez. VII (N.S.) 30 (1984) 27–40.

[B1] H. Bray, “The Penrose Inequality in General Relativity and Volume Comparison Theorems Involving Scalar Curvature,” thesis, Stanford University, 1997.

[B2] H. Bray, “Proof of the Riemannian Penrose Inequality using the Positive Mass Theorem,” J. Diff. Geom., to appear.

[BK] H. Brézis and D. Kinderlehrer, “The Smoothness of Solutions to Nonlinear Variational

Inequalities,” Indiana Univ. Math. J. 23 (1974) 831–844.

[CGG] Y.-G. Chen, Y. Giga, and S. Goto, “Uniqueness and Existence of Viscosity Solutions

of Generalized Mean Curvature Flow Equations,” J. Diff. Geom. 33 (1991) 749–786.

[C] Y. Choquet-Bruhat, “Positive-Energy Theorems,” Relativité, Groupes et Topologie II (Les Houches, 1983), ed. B. S. DeWitt and R. Stora, North Holland, Amsterdam, 1984, 739–785.

[CK] D. Christodoulou and S. Klainerman, “The Global Nonlinear Stability of Minkowski Space,” Princeton Math. Series #41, Princ. Univ. Press, Princeton, 1993.

[CY] D. Christodoulou and S.-T. Yau, “Some Remarks on the Quasi-Local Mass,” Mathematics and General Relativity (Santa Cruz CA 1986), ed. J. Isenberg, Contemporary Mathematics #71, AMS, 1988, 9–14.

[Ch1] P. T. Chruściel, thesis, Institute of Mathematics, Polish Academy of Sciences, 1985.

[Ch2] P. T. Chruściel, “Sur les Feuilletages ‘Conformement Minimaux’ des Varietés Riemanniennes de Dimension Trois,” C. R. Acad. Sci. Paris Ser. I Math. 301 (1985) 609–612.

[Ch3] P. T. Chruściel, “Boundary Conditions at Spatial Infinity from a Hamiltonian Point of View,” Topological Properties and Global Structure of Space-Time (Erice, 1985), NATO Adv. Sci. Inst. Ser. B: Phys. #138, Plenum, New York, 1986, 49–59.

[Ch4] P. T. Chruściel, “On Kijowski’s Unconstrained Degrees of Freedom of the Gravitational Field,” Proc. of the IV Marcel Grossman Meeting on General Relativity (Rome, 1985), North Holland, New York, 1986, 1643–1649.

[Ch5] P. T. Chruściel, “A Remark on the Positive Energy Theorem,” Class. Quan. Grav. 3 (1986) L115–L121.

[Ch6] P. T. Chruściel, “Sur les Coordonnées p-Harmoniques en Relativité Générale,” C. R. Acad. Sci. Paris Ser. I Math. 305 (1987) 797–800.

[CIL] M. G. Crandall, H. Ishii, and P.-L. Lions, “User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations,” Bull. of the Amer. Math. Soc. 27 (1992) 1–67.

[E] M. Emmer, “Superfici di Curvatura Media Assegnata con Ostacolo,” Ann. Mat. Pura Appl. 109 (1976) 371–389.

[ES] L. C. Evans and J. Spruck, “Motion of Level Sets by Mean Curvature I,” J. Diff. Geom. 33 (1991) 635–681.

[F] H. Friedrich, “Hyperbolic Reductions for Einstein’s Equations,” Class. Quant. Grav. 13 (1996) 1451–69.

[Ga] G. J. Galloway, “On the Topology of Black Holes” Comm. Math. Phys. 151 (1993) 53–66.

[Ge] C. Gerhardt, “Flow of Nonconvex Hypersurfaces into Spheres,” J. Diff. Geom. 32 (1990) 299–314.

[G] R. Geroch, “Energy Extraction,” Ann. New York Acad. Sci. 224 (1973) 108–17.

[G1] G. W. Gibbons, thesis, Cambridge University, 1973.

[G2] G. W. Gibbons, “The Time Symmetric Initial Value Problem for Black Holes,” Comm. Math. Phys. 27 (1972) 87–102.

[G3] G. W. Gibbons, “The Isoperimetric and Bogomolny Inequalities for Black Holes,” Global Riemannian Geometry, T. Willmore and N. Hitchin eds., Ellis Horwood, 1984.

[G4] G. W. Gibbons, “Collapsing Shells and the Isoperimetric Inequality for Black Holes,” Class. Quant. Grav. 14 (1997) 2905–2915.

[GT] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer, New York, 1983.

[GL1] M. Gromov and H. B. Lawson, “Spin and Scalar Curvature in the Presence of a Fundamental Group I,” Ann. of Math. 111 (1980) 209–230.

[GL2] M. Gromov and H. B. Lawson, “The Classification of Simply Connected Manifolds of Positive Scalar Curvature,” Annals of Math. 111 (1980) 423–434.

[GL3] M. Gromov and H. B. Lawson, “Positive Scalar Curvature and the Dirac Operator on Complete Riemannian Manifolds,” Inst. Hautes Études Sci. Publ. Math. #58, 1983, 83–196.

[HZ] R. Hardt and X. Zhou, “An Evolution Problem for Linear Growth Functionals,” Comm. Part. Diff. Eq. 19 (1994) 1879–1907.

[H1] S. W. Hawking, “Gravitational Radiation from Colliding Black Holes,” Phys. Rev.

Lett. 26 (1971) 1344.

[H2] S. W. Hawking, “Gravitational Radiation in an Expanding Universe,” J. Math. Phys. 9 (1968) 598–604.

[H3] S. W. Hawking, “Black Holes in General Relativity,” Comm. Math. Phys. 25 (1972) 152–166.

[HE] S. W. Hawking and G.F.R. Ellis, The Large-Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics, Cambridge, 1973.

[HP] S. W. Hawking and R. Penrose, “The Singularities of Gravitational Collapse and Cosmology,” Proc. Roy. Soc. A 314 (1970) 529–548.

[Ha] S. A. Hayward, “General Laws of Black-Hole Dynamics,” Phys. Rev. D 49 (1994) 6467-6474.

[Hd] M. Heidusch, “Higher Regularity of Inverse Mean Curvature Flow,” preprint, Tübingen, 2001.

[Hm] J. Hempel, 3-Manifolds, Princ. Univ. Press, Princeton, 1976.

[He] M. Herzlich, “A Penrose-like Inequality for the Mass of Riemannian Asymptotically Flat Manifolds,” Comm. Math. Phys. 188 (1997) 121–133.

[Ho] M. Holder, “Contracting Spacelike Hypersurfaces by their Inverse Mean Curvature,” J. Austral. Math. Soc. Ser. A 68 (2000) 285–300.

[Hu] G. Huisken, “Contracting Convex Hypersurfaces in Riemannian Manifolds by their Mean Curvature,” Invent. Math. 84 (1986) 463–480.

[HI1] G. Huisken and T. Ilmanen, “The Riemannian Penrose Inequality,” Int. Math. Res. Not. 20 (1997) 1045–1058.

[HI2] G. Huisken and T. Ilmanen, “A Note on Inverse Mean Curvature Flow,” Proc. of the Workshop on Nonlinear Part. Diff. Equ. (Saitama Univ., Sept. 1997), Saitama Univ., 1997.

[HY] G. Huisken and S.-T. Yau, “Definition of Center of Mass for Isolated Physical Systems and Unique Foliations by Stable Spheres with Constant Mean Curvature,” Invent. Math. 124 (1996) 281–311.

[Hut] J. E. Hutchinson, “Second Fundamental Form for Varifolds and Existence of Surfaces

Minimizing Curvature,” Indiana J. Math. 35 (1986) 45–71.

[I1] T. Ilmanen, Elliptic Regularization and Partial Regularity for Motion by Mean Curvature, Mem. Amer. Math. Soc. #520, Providence, 1994.

[I2] T. Ilmanen, “Lectures on Mean Curvature Flow and Related Equations,” lecture notes, ICTP, Trieste, 1995.

[J1] P. S. Jang, “On the Positive Energy Conjecture,” J. Math. Phys. 17 (1976) 141–145.

[J2] P. S. Jang, “On the Positivity of Energy in General Relativity,” J. Math. Phys. 19 (1978) 1152–1155.

[J3] P. S. Jang, “Note on Cosmic Censorship,” Phys. Rev. D 20 (1979) 834–838.

[J4] P. S. Jang, “On the Positivity of the Mass for Black Hole Space-Times,” Comm. Math. Phys. 69 (1979) 257–266.

[JW] P. S. Jang and R. M. Wald, “The Positive Energy Conjecture and the Cosmic Censor Hypothesis,” J. Math. Phys. 18 (1977) 41–44.

[Je1] J. Jezierski, “Positivity of Mass for Certain Spacetimes with Horizons,” Class. Quant.

Grav. 6 (1989) 1535–1539.

[Je2] J. Jezierski, “Stability of Reissner–Nordström Solution with Respect to Small Perturbations of Initial Data,” Class. Quant. Grav. 11 (1994) 1055–1068.

[Je3] J. Jezierski, “Perturbation of Initial Data for Spherically Symmetric Charged Black Hole and Penrose Conjecture,” Acta Phys. Pol. B 25 (1994) 1413–17.

[JK1] J. Jezierski and J. Kijowski, “A Simple Argument for Positivity of Gravitational Energy,” Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics (Clausthal, 1986), World Sci. Pub., Teaneck NJ, 1987, 187–194.

[JK2] J. Jezierski and J. Kijowski, “Positivity of Total Energy in General Relativity,” Phys. Rev. D 36 (1987) 1041–1044.

[K1] J. Kijowski, “Unconstrained Degrees of Freedom of Gravitational Field and the Positivity of Gravitational Energy,” Gravitation, Geometry and Relativistic Physics (Aussois, 1984), Lect. Notes in Phys. #212, Springer, Berlin, 1984, 40–50.

[K2] J. Kijowski, “On Positivity of Gravitational Energy,” Proc. of the IV Marcel Grossman Meeting on General Relativity (Rome, 1985), ed. R. Ruffini, North-Holland, Amsterdam, 1986, 1681–1686.

[LP] J. M. Lee and T. H. Parker, “The Yamabe Problem,” Bull. Amer. Math. Soc. 17 (1987) 37–91.

[Lc] A. Lichnerowicz, “Spineures Harmoniques,” C.R. Acad. Sci. Paris Ser. A-B 257 (1963) 7–9.

[LT] A. Lichnewsky and R. Temam, “Pseudosolutions of the Time-Dependent Minimal Surface Problem,” J. Diff. Eq. 30 (1978) 340–364.

[Ln] F.-H. Lin, “Regularity for a Class of Parametric Obstacle Problems,” thesis, Univ. Minnesota, 1985.

[Lo] J. Lohkamp, “Scalar Curvature and Hammocks,” Math. Ann. 313 (1999) 385–407.

[Lu] S. Luckhaus, “Solutions for the Two-Phase Stefan Problem with the Gibbs-Thompson Law for the Melting Temperature,” Eur. J. Appl. Math. 1 (1990) 101–111.

[Ma] U. Massari, “Esistenza e Regolarità delle Ipersuperfici di Curvatura Media Assegnata

in R^{n},” Arch. Rat. Mech. & Anal. 55 (1974) 357–382.

[MSY] W. H. Meeks III, L. M. Simon, and S.-T. Yau, “Embedded Minimal Surfaces, Exotic Spheres, and Manifolds with Positive Ricci Curvature,” Ann. of Math. 116 (1982) 621–659.

[MY] W. H. Meeks III and S.-T. Yau, “Topology of Three Dimensional Manifolds and the Embedding Problems in Minimal Surface Theory,” Ann. of Math. 112 (1980) 441–484.

[Mi] M. Miranda, “Frontiere Minimali con Ostacoli,” Ann. Univ. Ferrara Sez. VII (N.S.) 16 (1971) 29–37.

[PT] T. H. Parker and C. H. Taubes, “On Witten’s Proof of the Positive Energy Theorem,” Comm. Math. Phys. 84 (1982) 223–238.

[Pa] E. Pasch, “Numerische Verfahren zur Berechnung von Krümmungsflüssen,” Ph.D. thesis, Tübingen Univ., 1998.

[P1] R. Penrose, “Gravitational Collapse: The Role of General Relativity,” Rivista del Nuovo Cimento Numero Speciale 1 (1969) 252–276.

[P2] R. Penrose, “Naked Singularities,” Ann. New York Acad. Sci. 224 (1973) 125–134.

[P3] R. Penrose, “Some Unsolved Problems in Classical General Relativity,” Seminar on Differential Geometry, ed. S.-T. Yau, Annals of Math. Stud. #102, Princ. Univ. Press, Princeton, 1982, 631–668.

[PSW] R. Penrose, R.D. Sorkin, and E. Woolgar, “A Positive Mass Theorem based on Fo-

cussing and Retardation of Null Geodesics,” submitted to Comm. Math. Phys.

[R] O. Reula, “Existence Theorems for Solutions of Witten’s Equations and Nonnegativity of Total Mass,” J. Math. Phys. 23 (1982) 810–814.

[RT] O. Reula and K. P. Tod, “Positivity of the Bondi Energy,” J. Math. Phys. 25 (1984) 1004–1008.

[SSY] R. Schoen, L. M. Simon, and S.-T. Yau, “Curvature Estimates for Minimal Hypersurfaces,” Acta Math. 134 (1975) 275–288.

[SY1] R. Schoen and S.-T. Yau, “Incompressible Minimal Surfaces, Three-Dimensional Manifolds with Nonnegative Scalar Curvature, and the Positive Mass Conjecture in General Relativity,” Proc. Nat. Acad. Sci. 75 (1978) 45–76.

[SY2] R. Schoen and S.-T. Yau, “Complete Manifolds with Nonnegative Scalar Curvature and the Positive Action Conjecture in General Relativity,” Proc. Nat. Acad. Sci. 75 (1978) 45–76.

[SY3] R. Schoen and S.-T. Yau, “On the Proof of the Positive Mass Conjecture in General Relativity,” Comm. Math. Phys. 65 (1979) 45–76.

[SY4] R. Schoen and S.-T. Yau, “Positivity of the Total Mass of a General Space-Time,” Phys. Rev. Lett. 43 (1979) 1457–1459.

[SY5] R. Schoen and S.-T. Yau, “On the Structure of Manifolds with Positive Scalar Curvature,” Manuscripta Math. 28 (1979) 159–183.

[SY6] R. Schoen and S.-T. Yau, “Existence of Incompressible Minimal Surfaces and the Topology of Three Dimensional Manifolds with Non-Negative Scalar Curvature,” Ann. Math. 110 (1979) 127–142.

[SY7] R. Schoen and S.-T. Yau, “The Energy and the Linear Momentum of Space-Times in General Relativity,” Comm. Math. Phys. 79 (1981) 47–51.

[SY8] R. Schoen and S.-T. Yau, “Proof of the Positive Mass Theorem II,” Comm. Math. Phys. 79 (1981) 231–260.

[SY9] R. Schoen and S.-T. Yau, “Proof that the Bondi Mass is Positive,” Phys. Rev. Lett. 48 (1982) 369–371.

[SY10] R. Schoen and S.-T. Yau, “The Existence of a Black Hole due to Condensation of Matter,” Comm. Math. Phys. 90 (1983) 575–579.

[SY11] R. Schoen and S.-T. Yau, “The Structure of Manifolds with Positive Scalar Curvature,” Directions in Partial Differential Equations (Madison, 1985), Publ. Math. Res. Center Univ. Wisconsin #54, Academic Press, Boston, 1987, 235–242.

[S] L. M. Simon, Lectures on Geometric Measure Theory, Proc. of the Centre for Math.

Analysis #3, Austr. Nat. Univ., 1983.

[SWZ] P. Sternberg, G. Williams, and W. P. Ziemer, “C^{1,1}-Regularity of Constrained Area

Minimizing Hypersurfaces,” J. Diff. Eq. 94 (1991) 83–94.

[T1] I. Tamanini, “Boundaries of Caccioppoli Sets with Holder-Continuous Normal Vec-

tor,” J. Reine Angew. Math. 334 (1982) 27–39.

[T2] I. Tamanini, “Regularity Results for Almost Minimal Oriented Hypersurfaces in R^{n},” Quaderni del Dipartimento di Matematica dell’Università di Lecce 1 (1984) 1–92.

[To] K. P. Tod, “The Hoop Conjecture and the Gibbons-Penrose Construction of Trapped Surfaces,” Class. Quant. Grav. 9 (1992) 1581–1591.

[Tr] N. S. Trudinger, “Isoperimetric Inequalities for Quermassintegrals,” Ann. Inst. H. Poincaré. Analyse Non-Linéare 11 (1994) 411–425.

[U] J. Urbas, “On the Expansion of Starshaped Hypersurfaces by Symmetric Functions of Their Principal Curvatures,” Math. Z. 205 (1990) 355–372.

[V] A. Visintin, “Nucleation and Mean Curvature Flow,” Comm. Part. Diff. Eq. 23 (1998)

17–53.

[Wa1] R. M. Wald, General Relativity, Univ. Chicago Press, Chicago, 1984.

[Wa2] R. M. Wald, “Gravitational Collapse and Cosmic Censorship,” Black Holes, Gravitational Radiation and the Universe, Fund. Theories Phys. #100, Kluwer Acad. Publ., Dordrecht, 1999, 69–85.

[Wh] B. White, “Partial Regularity of Mean Convex Hypersurfaces Flowing by Mean Curvature,” Int. Math. Res. Not. 4 (1994) 185–192.

[Wi] T. Willmore, Total Curvature in Riemannian Geometry, Wiley, New York, 1982.

[W] E. Witten, “A New Proof of the Positive Energy Theorem,” Comm. Math. Phys. 80 (1981) 381–402.

[Y] J. W. York, Jr., “Kinematics and Dynamics of General Relativity,” Sources of Gravitational Radiation, ed. L. Smarr, Camb. Univ. Press, 1979, 83–126.

[Z] X. Zhou, “An Evolution Problem for Plastic Antiplanar Shear,” Appl. Math. Opt. 25 263–285.