The Riemannian Penrose Inequality

G. Huisken and T. Ilmanen

In this note we announce the proof of the Penrose inequality [P] for a Riemannian 3-manifold M^{3} of nonnegative scalar curvature, which gives a relation between the ADM mass of an end of the manifold, and the area of each “outermost” minimal sphere bounding the end. We employ the inverse mean curvature flow put forward by Geroch [G] and Jang-Wald [JW]. The complete proof will appear in [HI].

The result has a certain interpretation in general relativity. The Riemannian manifold M can arise as a maximal spacelike hypersurface in an asymptotically flat Lorentzian manifold L^{3,1} modelling a relativistic isolated gravitating system. In this setting, the physical assumption of nonnegative local energy density, expressed by the “weak energy condition”, implies in conjunction with Einstein’s equations and the Gauss equation that

R ≥ 0

where R is the scalar curvature of M in its induced metric. The Penrose inequality then establishes an optimal lower bound for the total energy of the isolated system measured at spacelike infinity, in terms of the size of a black hole contained in the system, measured by the area of its apparent horizon.

An end of M is called asymptotically flat if it is diffeomeorphic to the complement of a compact set K in R^{3}, and the metric tensor g = (g_{ij}) of M satisfies

|g_{ij} − δ_{ij}| ≤ C/|x|,  |g_{ij,k}| ≤ C/|x|^{2},  Rc ≥ −Cδ/|x|^{2},

as |x| → ∞, where the derivatives are taken with respect to the Euclidean metric δ = (δ_{ij}) on R^{3} \ K, and the scalar curvature of g satisfies

∫_{M} |R| < ∞.

See [CK, Y, F] for a discussion of asymptotic flatness.

The ADM mass of the end (see [ADM, Y]) is defined by

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

Although expressed in coordinates, this expression yields a geometric invariant of the asymptotic regime of M, see [Ch], [Ba2]. In the case that g is rotationally symmetric up to first order, g will have the form

g_{ij} = δ_{ij} (1 + \frac{m}{2r} )^{4} + O (\frac{1}{r^{2}} ) .

The ADM mass measures the gravity flux through a large celestial sphere.

The Riemannian case of the Positive Mass Theorem, proven by Schoen-Yau [SY3], states the following. A minimal surface shall mean a smooth surface with vanishing mean curvature.

Let M be a 3-manifold with asymptotically flat ends, whose boundary is a compact minimal surface or empty. If M has nonnegative scalar curvature, then each end of M satisfies

m ≥ 0

with equality only when M is flat Euclidean space.

The full version of the positive mass theorem [SY6] treats general asymptotically flat initial data sets, and replaces the hypothesis R ≥ 0 by a more complicated expression involving also the extrinsic curvature data of M inside the Lorentzian manifold L. There is a generalization to higher dimensions as well, see [SY2, SY9].

The positive mass theorem says, roughly speaking, first that the positive local mass density (corresponding to R) adds up, in some indirect way, to yield positive total mass; and second, that a given amount of matter cannot compress into a volume smaller than its own Schwarzschild radius (except by retreating inaccessibly behind a black hole), so the magnitude of the (negative) gravitational potential energy cannot exceed the mass-energy of the matter fields, including kinetic energy. Such a limitation does not hold in Newtonian gravitational mechanics. In general relativity, these quantities cannot be separated, thus the difficulty of this theorem. See Geroch [G].

The positive mass theorem provides a consistency check on general relativity, and indicates the stability of Minkowski space (and hence of regular points of any spacetime) against energy-decreasing decay. See Witten [W], who gives another proof using spinors; also see [R]. Other applications of spinors to positive scalar curvature originate in Lichnerowicz [Li], Gromov-Lawson [GL1, GL2, GL3]. For the proof of the stability of Minkowski space under the Einstein evolution equation, see the recent book of Christodoulou-Klainerman [CK].

Let us turn to the role of 2-surfaces. A marginally trapped surface is a closed surface N^{2} in L whose mean curvature vector H̃ in L is light-like and past-directed. This implies that both inward and outward future-directed light rays emanating orthogonally from N are convergent in the mean. By the Hawking-Penrose singularity theorems [HE, HP], under general conditions the existence of N forces a singularity to form later in time. According to the (weak) Cosmic Censorship hypothesis, for generic data an event horizon forms around the trapped surface and the singularity, preventing light or other information from escaping to infinity. For this reason, N is called an apparent horizon.

If the extrinsic curvature of M in L happens to vanish along N, then N is a minimal surface in M. This brings us to the purely Riemannian setting for our theorem.

We call M an outermost region if M is a complete Riemannian 3-manifold with smooth, compact boundary, such that

(i) M is asymptotically flat with one end,

(ii) ∂M has vanishing mean curvature, and M contains no other compact minimal surfaces.

Remarks. 1. If M̃ is any 3-manifold with asymptotically flat ends and minimal, compact boundary, then each end is realized by a unique outermost region, and these outermost regions are disjoint.

2. An outermost region is diffeomorphic to R^{3} minus a finite number of disjoint open 3-balls, and each boundary component is a 2-sphere of vanishing mean curvature that minimizes area in its homology class. See White [Wh].

Main Theorem. Let M be an outermost region. Suppose that the scalar curvature of M is nonnegative. Then

|N|^{1/2} ≤ 4\sqrt{π} m,

where m is the ADM mass of M, and |N| is the area of any one of the minimal spheres bounding M. Equality holds if and only if M is one-half of the spatial Schwarzschild manifold.

Our theorem includes the positive mass theorem as a consequence.

The spatial Schwarzschild metric is given in conformal coordinates on R^{3} \ {0} by

(1 + \frac{m}{2r} )^{4} δ_{ij},

where δ_{ij} is the flat metric, r = |x| > 0, and m > 0. It has 2 asymptotically flat ends, a minimal sphere of area 16πm^{2} at r = m/2, and ADM mass m. Therefore, it realizes equality in the Penrose inequality. It is the unique complete, rotationally symmetric manifold of vanishing scalar curvature, and could be called the catenoid of intrinsic geometry (like the catenary, it is defined purely by the influence of gravity!). The Schwarzschild manifold is the initial data-set for the Schwarzschild space-time, which is the universal model of a static, nonspinning black hole in empty space. Our result is relevant to the stability of the Schwarzschild space-time.

Spatial Schwarzschild Manifold.

The “outermost” condition is essential. For example, one can construct a rotationally symmetric example by taking a chain of spheres connected by Schwarzschild-like minimal necks of various sizes, in such a way that the scalar curvature is still positive. See [SY5] or [GL1, GL2] for such a construction. The last neck, which blossoms out to half a Schwarzchild manifold, may have arbitrarily small area. It shields the others from view (literally).

Penrose deduced the inequality via a physical argument involving the Cosmic Censorship hypothesis (see [P], [JW] or [BI]), and proposed to disprove Cosmic Censorship by finding an initial data set that violates the inequality. Our theorem rules out this strategy. However, Christodoulou has recently found (nongeneric) naked singularities [C].

Related results have been obtained by Bray [B1, B2] using isoperimetric regions, by Herzlich [He] via the Dirac operator, Gibbons [Gi] in the special case of collapsing shells, as well as Tod [T], and Bartnik [Ba3] in the class of metrics admitting quasi-spherical foliations.

An important quantity in our work is the Hawking quasi-local mass (see [H1]) associated to a connected surface N in M according to the formula

m_{H}(N) = \frac{1}{64π^{3/2}} |N|^{1/2} (16π − ∫_{N} H^{2}) .

A principal motivation for this quantity is that the quasi-local mass of a large coordinate sphere approximates the ADM mass. One consequence of our work is that

m_{ADM}(M) = sup_{N} m_{H}(N)

where N ranges over all connected, compact surfaces in M that minimize area among surfaces separating N from infinity.

Another consequence is positivity of the Bartnik mass [Ba4] of a nonflat domain E of nonnegative scalar curvature

m_{B}(E) > 0

where m_{B}(E) is defined to be the infimum of the mass of asymptotically flat extensions of E with nonnegative scalar curvature and no horizons. A further consequence is the exhaustion property proposed by Bartnik, namely m_{B}(E) → m_{ADM}(M) as E → M.

Sketch of proof. Our proof employs the flow of surfaces by the inverse mean curvature −1/H, proposed by Geroch [G] and further developed by Jang-Wald [JW], Jang [J2]. Geroch puts forward a remarkable calculation that shows that the quasi-local mass is nondecreasing, and interpolates between the area of the minimal surface and the ADM mass [JW]. As long as the flow remains smooth, this proves the Penrose inequality.

We succeed in justifying their calculation, even when singularities and jumps occur in the moving surface. We use a minimizing principle for the weak evolution to select the correct flow; it also limits the topology of the moving surface, and conveniently jumps over the parts of the manifold that are shielded by minimal surfaces. Our variational formulation is inspired by Hardt-Zhou [HZ], Luckhaus [L], and Visintin [V], who observed the possibility of jumps. Because the mean curvature is bounded, results of geometric measure theory allow us to control the surfaces in C^{1,α}. We prove existence of the flow by means of elliptic regularization, an approximation scheme with a small ε. We perform the Geroch calculation at the ε level, and pass to limits to prove the theorem.

We will first discuss the case when M has only a single boundary component N_{0}. Consider the following parabolic evolution equation for a family of surfaces {N_{t}}_{t≥0} starting at N_{0},

(⋆)  \frac{∂}{∂t}x = \frac{1}{H} ν,  x ∈ N_{t}, t ≥ 0,

where ∂x/∂t is the normal velocity of the surface, ν is the outward unit normal, H is the mean curvature of N_{t} in M defined by H = −H⃗ · ν, and H⃗ is the mean curvature vector of N_{t}.

An example is a sphere in R^{3}, which remains round and expands exponentially fast. If N_{0} is a smooth, starshaped hypersurface of positive mean curvature in R^{n}, then N_{t} exists forever and converges to an expanding round sphere, as shown by Gerhardt [Ge]. See also Urbas [U], and [Ho] for work in Lorentzian manifolds.

We derive the following evolution equations,

(2)  \frac{d}{dt}|N_{t}| = |N_{t}|,  so  |N_{t}| = e^{t}|N_{0}|,

and for the mean curvature,

\frac{d}{dt}H = Δ (−\frac{1}{H} ) − \frac{|A|^{2}}{H} − \frac{Rc(ν, ν)}{H}.

where Rc is the ambient Ricci curvature. By the maximum principle, the latter equation implies that

sup_{[0,T]} sup_{N_{t}} |H| ≤ C(T)

for each T < ∞, as long as H remains positive. This important estimate remains true in the weak setting.

Let m(t) be the quasi-local mass of N_{t}. Clearly

m(0) = \frac{1}{4\sqrt{π}} |N|^{1/2}.

As mentioned above, if N_{t} becomes a large, approximately round sphere, then we expect that

m(t) → m_{ADM}(M).

As long as the flow remains smooth, a calculation using the Gauss formula and Gauss-Bonnet formula [G] shows that

(3) \frac{d}{dt}m(t) = \frac{1}{64π^{3/2}} |N_{t}|^{1/2} [16π − 8πχ(N_{t}) + ∫_{N_{t}}(2|∇ log H|^{2} + (λ_{1} − λ_{2})^{2} + R) dμ] .

where χ(N_{t}) is the Euler characteristic, λ_{1}, λ_{2} are the principal curvatures of N_{t} in M, and R is the scalar curvature of M. (The calculation resembles that of Schoen-Yau [SY3] for stable minimal surfaces in manifolds of positive scalar curvature.)

Assuming that N_{t} is connected, we have χ(N_{t}) ≤ 2, and therefore

\frac{d}{dt}m(t) ≥ 0.

If it happens that the flow exists and remains smooth forever, then collecting together the above inequalities, we have

\frac{1}{4\sqrt{π}} |N|^{1/2} ≤ m_{ADM}(M),

proving the Main Theorem.

Difficulties. In order to make the above idea really work, we must overcome several obstacles:

(a) In general, the flow is not smooth. The intrinsic evolution equation for the length squared of the second fundamental form A = (A_{ij}) is given by

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

+ A ∗ A ∗ Rc + A ∗ DRc),

where ∇, Δ are derivatives on N_{t}, and A ∗ A ∗ Rc, A ∗ DRc represent contractions of A with ambient curvature quantities. This suggests that the second fundamental form blows up, particularly if H → 0. In fact, examples show that H does go to zero, and that N_{t} cannot be better than C^{1,1} in general.

(b) In order to preserve the desirable condition H > 0 (or H ≥ 0), it appears that N_{t} must move discontinuously. For example, a torus in the shape of a thin ring will inexorably fatten up to the point where it loses positive mean curvature, unless it jumps to a circumscribing sphere.

(c) Suppose that there are other minimal surfaces between N_{t} and the end. It seems that N_{t} cannot pass these barriers without developing infinite speed somewhere. How can we use the hypothesis that there are no other minimal surfaces in M?

We overcome these difficulties by introducing several ideas.

Level-set Ansatz (see [ES]). We assume that the surface always moves (weakly) in the same direction, so that H ≥ 0, and the evolution is represented by a function t = u(x) such that N_{t} = {u = t}. Then (⋆) becomes the following equation for u,

(⋆⋆)  H ≡ div (\frac{Du}{|Du|}) = |Du|,

where div, D are derivatives in M. Note that |Du| is the inverse speed. Define E_{t} = {u < t}, the region already crossed by N_{t}. For convenience, fill in N_{0} with a ball B_{0} and set E_{0} = B_{0}.

Minimization Principle. Freezing the |Du| term, equation (⋆) is the Euler-Lagrange equation of the functional

(J)  J_{u}(E) = |∂E| − ∫_{E} |Du| dx,

where E is a set of finite perimeter containing N_{0}. By the co-area formula, this is equivalent to the functional

(J′)  J_{u}(v) = ∫_{M} |Dv| + v|Du| dx,

where v is a locally Lipschitz function on M with v = 0 on N_{0}. (The local Lipschitz property corresponds to the local boundedness of H, mentioned above.) We require that u minimize J_{u}(v) with respect to compactly supported variations, or equivalently, that each E_{t} is a minimizer of J_{u}(E) among finite perimeter sets containing N_{0}.

Interestingly, the Geroch monotonicity (3) is the stability inequality for the functional J_{u}(E), applied in the neutral direction ∂/∂t.

Heuristic Argument. The minimization principle serves as a selection mechanism that tells the surfaces “when to jump”. The key idea is contained in the following heuristic description of the flow.

For any bounded open set E, define the (maximal) mean convex hull E′ to be the unique open set of largest volume among all least-perimeter sets containing E.

Since M contains no minimal surfaces besides its boundary, E′_{0} = E_{0}. We flow E_{t} by (⋆) as long as E_{t} remains equal to E′_{t}. Whenever this condition is violated, E_{t} jumps to E′_{t}, and the flow continues.

For example, if E_{t} is the union of two disjoint, expanding balls in R^{3}, then E_{t} coalesces to a single lump as soon as there is an enclosing connected set of perimeter equal to the combined area of the two spheres.

Jump time.

At a jump, we necessarily have |∂E_{t}| = |∂E′_{t}|, so (2) continues to hold. On the other hand, ∂E′_{t} solves an obstacle problem enclosing E_{t}, so the mean curvature of ∂E′_{t} is zero where there is no contact, and equal to that of ∂E_{t} on the contact set. Therefore, formally at least,

∫_{∂E′_{t}} H^{2} ≤ ∫_{∂E_{t}} H^{2}

from which it follows that m(t) is continuous or experiences a jump increase, consistent with the Geroch monotonicity (3).

There would be several difficulties in making the above sketch into a proof. In our method, we do not deal directly with the jump times, but instead use the weak formulation and an approximation scheme to work more globally. We now turn to a more detailed description of what we actually do.

Compactness and Hölder Regularity. Consider the class of locally Lipschitz functions u that minimize J_{u}(v) with respect to compactly supported variations. We prove a compactness theorem for this class. Since the integrand |Du| in M_{u} is locally bounded, standard results in geometric measure theory imply that the level sets of u converge locally uniformly in C^{1,α}. Using the Gauss-Bonnet formula, it follows that ∫_{N_{t}} |A|^{2} is (locally) bounded.

Topology. By a replacement argument, it is easy to verify that a function u that minimizes J_{u}(v) has no local maxima or minima. Since N_{0} is connected, and M is simply connected and connected at infinity, this implies that N_{t} remains connected. (We are grateful to P. Li for clarifying this point for us.) Ultimately, this is essential for the validity of the Geroch monotonicity dm/dt ≥ 0.

Elliptic Regularization (see [I]). In order to prove existence of minimizers of (J′), we employ the approximation

(⋆)_{ε}  div (\frac{Du^{ε}}{\sqrt{|Du^{ε}|^{2} + ε^{2}}} ) = \sqrt{|Du^{ε}|^{2} + ε^{2}}.

This equation has a geometric interpretation: it states that the function u^{ε}(x)/ε − t/ε solves the inverse mean curvature flow (⋆⋆) in the manifold M × R. That is, the graph of v^{ε} := u^{ε}/ε is a translating solution of (⋆) in M × R, with downward speed 1/ε.

By constructing sub- and supersolutions, we show that solutions of (⋆)_{ε} with boundary N_{0} exist over a ball of radius C/ε in M. Passing the graphs of v^{ε_{i}} − t/ε_{i} to limits using the compactness theorem above, we obtain a family of cylinders in M × R that solve (⋆), which we slice by M to obtain global solutions of (⋆) in M that minimize (J).

Evolution of the Quasi-Local Mass. Returning to the approximate equation and inserting a suitable test function, we obtain a version of the Geroch inequality at the ε level. There are some new terms, but they all have the correct sign or vanish in the limit.

To prove the Theorem, we must pass ε → 0. This involves weak limits of curvature quantities such as

∫_{graph v^{ε}} H^{2},  ∫_{graph v^{ε}} |A|^{2},  ∫_{graph v^{ε}} \frac{|∇H|^{2}}{H^{2}} ,

integrated over a family of 3-manifolds in M × R that are converging locally in C^{1,α} and in W^{2,2}.

The first integral has pointwise a.e. convergence by the Rellich theorem and the integral estimate on ∇H coming from (3) and the fact that |H| is bounded. The second two integrals are lower semicontinuous. In the limit, the connectedness of the N_{t} comes into play and we can prove

m(s) ≤ m(t) for all 0 < s ≤ t.

Initial Layer. If there were a surface of smaller area than N_{0} separating N_{0} from infinity, then N_{0} would immediately jump to this surface, with a drop in area, and the Geroch monotonicity would be lost. The assumption that M is an outermost region rules this out, however, and shows that

m(0) ≤ m(t)

for all t ≥ 0.

Long-Term Convergence. By spatial blowdown to the standard flow on Euclidean space, we establish that for large t, N_{t} is C^{1,α} close to a large sphere, relative to the radius. We then employ the conformal test vectorfield x/|x|^{2} of Trudinger [T] to prove that the Hawking mass is approximately bounded by the flux integral in (1). Passing t → ∞, this proves that lim_{t→∞} m(t) ≤ m_{ADM}(M).

Other Boundary Components. Let us now suppose that M has boundary spheres N_{0}, N_{1}, …, N_{k}. We extend M by filling each N_{i} with a ball B_{i}. Let E_{t} be the evolution of B_{0} in the extended manifold, and let F_{t} = E′_{t} ∪ B_{1} ∪ ⋯ B_{k}, where E′_{t} is the maximal mean convex hull of E_{t}. We continue the evolution of E_{t} until F_{t} ceases to be equal to F′_{t}. At that instant, F′_{t} has the same perimeter as F_{t}, but the connected component E_{t}^{+} of F′_{t} containing E_{t} also contains at least one of the balls B_{i}, i ≠ 0. By considering the obstacle problem as in theAs in the heuristic argument for jump times, we find that m_{H}(E_{t}) ≤ m_{H}(E′_{t}). Continuing the evolution from this component, eventually all the balls are engulfed, and the Penrose inequality follows.

Case of Equality. If m(t) remains constant for any interval, then by (3) we find that for a.e. t, H is a bounded constant H(t) and A = H(t)g/2, from which it follows that N_{t} is C^{2}. By continuity, these properties hold for every t in the interval. It follows that there are no jumps, unless one of the N_{t} is a minimal surface, which is ruled out except for N_{0}. Therefore in this interval, the surfaces N_{t} form a smooth foliation by level sets of a distance function. Employing the flow equation, we deduce that each N_{t} is isometric to a round 2-sphere, and we obtain an ODE for H(t), from which it follows that M is locally isometric to the Schwarzschild metric. Since there is no occasion for N_{t} to jump, M is

equal to half of the Schwarzschild manifold.

Acknowledgements. We would like to thank B. Andrews, G. Belletini, R. Hamilton, P.

Li, R. Schoen, and S.-T. Yau for stimulating and useful conversation. The paper started

at the CIME meeting in Cetraro and the PDG meeting in Oberwolfach, in summer 1996.

Questions.

1. What about an additive combination of the areas of several boundary components?

2. Develop a space-time version of the theorem (drop the requirement that M is

totally geodesic in L).

3. Generalize the theorem to higher dimensions.

4. What is the relation to the Witten spinor proof?

5. Prove that the inverse mean curvature flow is unique, and is smooth except at the

jump times, when it is C^{1,1}. If M is analytic, then the number of jump times is finite.

Gerhard Huisken Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 D-72076 Tübingen Germany Tom Ilmanen Max-Planck Institut für Mathematik 22-26 Inselstrasse, Stock 3 D-04103 Leipzig Germany

References

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

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

[Ba4] R. Bartnik, “New Definition of Quasi-Local Mass,” Phys. Rev. Lett. 62 (1989) 2346.

[BI] C. Barrabès and W. Israel, Phys. Rev. Lett. D 43 (1990) 1129–1142.

[B1] H. Bray, thesis, Stanford University, 1997.

[B2] H. Bray, research announcement, 1997.

[C] D. Christodoulou, “Examples of Naked Singularity Formation in the Gravitational Collapse of a Scalar Field,” Ann. of Math. 140 (1994) 607-653.

[CK] D. Christodoulou and S. Klainerman, “The Global Nonlinear Stability of Minkowski Space,” Princeton Math. Series v. 41, Princeton 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.

[Ch] P. 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, 49-59.

[E] L.C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Reg. Conf. Ser. #74, Amer. Math. Soc., Providence.

[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. Quantum Grav. 13 (1996) 1451–69.

[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.

[Gi] G.W. Gibbons, “Collapsing Shells and the Isoperimetric Inequality for Black Holes,” Univ. of Camb. Preprint hep-th/9701049, preprint, 1997.

[GG] G. Gibbons, “The Isoperimetric and Bogomolny Inequalities for Black Holes,” Global Riemannian Geometry, T. Willmore and N. Hitchin eds., Ellis Horwood, 1984.

[GL1] M. Gromov and H.B. Lawson, “Spin and Scalar Curvature in the Presence of a Fundamental Group I,” Annals 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 Etudes Sci. Publ. Math. No. 58, 1983, 83–196.

[HZ] R. Hardt and X. Zhou, “An Evolution Problem for Linear Growth Functionals,” Comm. Part. Diff. Equ. 19 (1994) 1879–1907.

[H1] S. W. Hawking, “Gravitational Radiation in an Expanding Universe,” J. Math. Phys. 9 (1968) 598-604.

[H2] 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, 1973.

[HP] S. W. Hawking and R. Penrose, “The Singularities of Gravitational Collapse and Cosmology,” Proc. Roy. Soc. A 314 (1970) 529-548.

[He] M. Herzlich, “A Penrose-like Inequality for the Mass of Riemannian Asymptotically Flat Manifolds,” Prepublications du Centre de Math. Ecole Polytechnique #1146 (1996), to appear in Comm. Math. Phys.

[Ho] M. Holder, “Contracting Spacelike Hypersurfaces by their Inverse Mean Curvature,” Preprint SFB 382 Tübingen, No.57 (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.

[HI] G. Huisken and T. Ilmanen, “The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality,” in preparation.

[I] T. Ilmanen, Elliptic Regularization and Partial Regularity for Motion by Mean Curvature, Mem. Amer. Math. Soc. #520, 1994.

[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. Lett. 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.

[Li] A. Lichnerowicz, “Spineures Harmoniques,” C.R. Acad. Sci. Paris Ser. A-B 257 (1963) 7-9.

[L] 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.

[P] R. Penrose, “Naked Singularities,” Ann. New York Acad. Sci. 224 (1973) 125-134.

[R] O. Reula, “Existence Theorems for Solutions of Witten’s Equations and Nonnegativity

of Total Mass,” J. Math. Phys. 23 (1982) 810–814.

[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 Manifold 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, “Proof of the Positive Mass Theorem II,” Comm. Math. Phys. 79 (1981) 231–260.

[SY7] R. Schoen and S.-T. Yau, “The Energy and the Linear Momentum of Space-Times in General Relativity,” Comm. Math. Phys. 81 (1981) 47–51.

[SY8] R. Schoen and S.-T. Yau, “The Existence of a Black Hole due to Condensation of Matter,” Comm. Math. Phys. 90 (1983) 575–579.

[SY9] R. Schoen and S.-T. Yau, “The Structure of Manifolds with Positive Scalar Curvature,” Directions in Partial Differential Equations, Madison, 1985, 235–242, Publ. Math. Res. Center Univ. Wisconsin #54, Academic Press, Boston, 1987.

[T] K. P. Tod, Class. Quant. Grav. 3 (1992) 1581–1591.

[Tr] N. S. Trudinger, “A Sharp Inequality for Subharmonic Functions on Two-Dimensional Manifolds,” Math. Z. 133 (1973) 75-79.

[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,” preprint.

[Wh] B. White, “On the Topological Type of Minimal Submanifolds,” Topology 31 (1992) 445–448.

[Wi] 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, p.83.