Elliptic Regularization and Partial Regularity for Motion by Mean Curvature

Tom Ilmanen University of Wisconsin Madison WI 53706

February, 1993

Contents

Introduction .......................................................................................vii Chapter I. Elliptic Regularization ........................................................1

§1 Varifolds and Currents §2 Translative Functional §3 Existence of Approximators §4 H^{2} Estimate §5 Space-Time Mass Bound §6 The Moving Varifolds of Brakke §7 Convergence Lemma for Integral Brakke Motions §8 Enhanced Motion §9 Matching Motion

Chapter II. Partial Regularity in Codimension One...........................55 §10 Avoidance and Inclusion for Brakke Motions §11 Unit Density §12 Almost Everywhere Regularity

Appendix............................................................................................83 References ..........................................................................................87

Abstract

I. We study Brakke’s motion of varifolds by mean curvature in the special case that the initial surface is an integral cycle, giving a new existence proof by mean of elliptic regularization. Under a uniqueness hypothesis, we obtain a weakly continuous family of currents solving Brakke’s motion.

II. These currents remain within the corresponding level-set motion by mean curvature, as defined by Evans-Spruck and Chen-Giga-Goto. Now let T_{0} be the reduced boundary of a bounded set of finite perimeter in R^{n}. If the level-set motion of the support of T_{0} does not develop positive Lebesgue measure, then there corresponds a unique integral n-current T, ∂T = T_{0}, whose time-slices form a unit density Brakke motion. Using Brakke’s Regularity Theorem, spt T is smooth H^{n}-almost everywhere. In consequence, almost every level-set of the level-set flow is smooth H^{n}-almost everywhere in space-time.

Subject Classifications: 35, 53, 82

Keywords: Mean Curvature, Phase Transitions, Level-Set Flow, Weak Convergence

Introduction

A surface moving so as to decrease its area most efficiently is said to be moving by mean curvature. In certain physical phenomena, for example, the energy of an interface between two phases of matter is proportional to the surface area. If momentum, pressure, and other forces can be ignored, then the interface is flowing by mean curvature.

From another point of view, the mean curvature flow is the only possible isotropic, homogeneous, quasilinear parabolic equation for the motion of a surface in space. As such it has great geometric significance and beauty and is closely related to such other geometric flows as Hamilton’s Ricci flow [Ha1, Ha2, Ha3], the harmonic map heat flow Chen [Ch], Chen-Struwe [CS], Struwe [St], and the Yang-Mills flow [Do].

Brakke initiated the study of the generalized mean curvature flow in his 1978 book [B], using methods of geometric measure theory. In the early eighties, Gage [Ga] and Gage-Hamilton [GH] showed that convex curves evolve smoothly, becoming asymptotically round as they disappear. Grayson extended the smoothness to general embedded curves in [Gr1, Gr2], solving a problem posed by Gauss. Angenent has shown how to evolve immersed curves in [An1, An2, An3]. Huisken established the asymptotic behavior of convex bodies in all dimensions in [H1], and studied general singularity formation in [H2]. Comprehensive estimates for graphs evolving by mean curvature appear in Ecker-Huisken [EH1, EH2]. These papers employ classical differential geometry and the maximum principle.

Allen and Cahn [AC] conjectured mean curvature motion as the singular limit of a reaction-diffusion (phase-field) equation. This idea has been developed by de Mottoni and Schatzman [MS1, MS2, MS3], Bronsard-Kohn [BK], and many others. Following Osher-Sethian’s numerical work [OS], Chen-Giga-Goto [CGG] and Evans-Spruck [ES1, ES2, ES3, ES4] introduced the level-set-flow, in which the moving surface is the zero-set of a function, all of whose level-sets move by mean curvature. The phase-field and levelset approaches are reconciled in the paper of Evans-Soner-Souganidis [ESS], and unified with Brakke’s work in Ilmanen [I3]. Other interesting papers along these lines include Altschuler-Angenent-Giga [AAG] and Barles-Soner-Souganidis [BSS].

In the work of Luckhaus [Lu], Almgren-Wang [AW], and Almgren-Taylor-Wang [ATW], the flow is approximated by a time-step minimization method that is broadly

Received by the editor January 3, 1992.

INTRODUCTION

related to the method of this paper. There are also stochastic approximations, discrete approximations, etc.

In this work we introduce a new approximation scheme for Brakke’s mean curvature flow of varifolds. Using geometric measure theory, we pass to limits and in the process sharpen and clarify his definition. Next, we synthesize Brakke’s flow with the level-set flow. Turning to maximum principle (viscosity) techniques, we succeed in partly closing an annoying gap in Brakke’s regularity theory, yielding (generic) almost-everywhere regularity for motion by mean curvature.

The reader should start with the introductions to Chapters One and Two, then proceed directly to §2, referring to §1 as needed.

—————————

I would like to thank Jacki Jae-Kyung K. Noh, Lisa Anne Rhode, and Edith Hooker Ilmanen for their faith. I am grateful for mathematical conversations over the years with Steven Altschuler, Sigurd Angenent, Andrew Casson, Yoshikazu Giga, Richard Hamilton, Michael D. Hirsch, Bruce Kleiner, Robert Kusner, Helena Lopes, Yong-Bin Ruan, H. Mete Soner, Takis Souganidis, Lang-Fang Wu, and Yu-Xi Zheng. Also I would like to thank A. Chorin for support from the National Science Foundation, and the Institute for Advanced Study for hospitality during the final preparation.

This work relies heavily on the ideas of Brakke in his 1978 book [B], which are only now beginning to be understood and applied.

My special thanks go to my adviser, L. C. Evans, whose arrival at Berkeley came at a critical moment for me. Our conversations and his stellar notes and classes have contributed greatly to my mathematical development. Without him this work would not have been possible.

Typeset by AMS-TEX.

Notation

M[T], M[μ] mass of current, Radon measure
Λ_{k}M^{n}  bundle of alternating k-vectors on M
Λ^{k}M^{n}  bundle of alternating k-forms on M
P_{A}  P ⌊M × A, A ⊆ R
P_{z}  ∂(P ⌊M × [z, ∞))
S(x)  approximate tangent k-plane of the Radon measure μ at the point x
S · v  projection of the vector v onto the k-plane S
S : T  inner product of linear transformations S and T
R^{+}  [0, ∞)
D̄f(t)  upper derivate, i.e. lim sup_{s→t} (f(s) − f(t))/(s − t)
D^{+}f(t)  right upper derivate, i.e. lim sup_{s↓t}(f(s) − f(t))/(s − t)
D^{−}f(t)  left upper derivate, i.e. lim inf_{s↑t}(f(s) − f(t))/(s − t)
∇, ¯∇, D, D̄  covariant derivatives
ω  ∂/∂z
|A|  L^{1}(A)

Conventional Symbols

Γ_{t}  level set
T, P  current
μ, ν  Radon measure
V  varifold
X  rectifiable set
ϕ, ψ  test function
Φ  diffeomorphism
Y  vector field
U  open set
u  function
M^{n}  Riemannian manifold of dimension n
g  metric

x

Chapter One. Elliptic Regularization

In order to model the flow of surfaces through singularities, Brakke [B] developed a theory of varifolds moving by mean curvature. It consists of three parts: definition, existence, and regularity.

Brakke’s theory suggests two general questions about motion by mean curvature. First, there is the task (see §6) of sharpening Brakke’s definition to resolve difficulties concerning the continuity of the motion and the attainment of initial conditions. In fact, Brakke’s moving varifolds seem to be (measure-theoretic) subsolutions, just as the levelsets of Evans-Spruck and Chen-Giga-Goto are (set-theoretic) supersolutions.

Second, there is the question of the lacuna between Brakke’s existence theory and his regularity theory. If the moving varifolds are of unit density, then Brakke proves regularity almost everywhere. However, his existence construction may not satisfy this hypothesis, even if the initial surface has unit density.

Brakke’s theory is valid for surfaces with junctions. If we restrict ourselves to the smaller class of orientable surfaces without boundary (but still possibly singular), then it is possible to provide an alternate existence proof for Brakke’s motion, by means of elliptic regularization. The basic idea is to add a small second derivative in the time direction.

This construction has several advantages. Speaking generally, it is elegant and suggestive, and might be useful in other parabolic problems of low regularity. In the problem at hand, it yields not only a family of integer rectifiable varifolds moving by mean curvature in the sense of Brakke, but also an integral current, T, in space-time, whose boundary is the initial surface.

If it happens that T is unique, then, as we prove in §9, the time-slices of T themselves form a family moving by mean curvature in the sense of Brakke.

Let us describe the construction in more detail. Suppose T_{0} is a locally integral current in M × {t = 0}, where M is a manifold. In §2, we introduce a functional I^{ε} defined by

I^{ε}[P] = \frac{1}{ε} ∫_{P} e^{−z/ε}dμ_{P} (x, z)

where (x, z) are coordinates for M × R and P is a locally integral (k + 1)-current whose boundary is the initial surface T_{0}. A stationary point P^{ε} for I^{ε} is a translating soliton 2 TOM ILMANEN

for motion by mean curvature. To be precise, if P^{ε} begins to evolve by mean curvature in M × R, P^{ε} will translate with velocity vector −(1/ε)(∂/∂z).

Let us write T^{ε} for P^{ε} rescaled by the transformation t = εz. The fundamental observation in §2 is that T^{ε} closely approximates the track in space-time of a motion by mean curvature. In §3 we prove the existence of these approximators T^{ε}. In the two following sections, we turn the fundamental estimate for the mean curvature flow, namely

Area(T_{0}) = Area(T_{t}) + ∫_{0}^{t} ∫_{T_{s}} H^{2} dμ_{s} ds,

into the estimates needed to pass to limits as ε ↓ 0.

Next, in §6, we define Brakke’s varifolds moving by mean curvature, and prove a compactness theorem for them in §7. In §8 we apply this theorem to T^{ε} to obtain, as ε ↓ 0, a pair

(T, {μ_{t}}_{t≥0}).

where T is a locally integral current in M × [0, ∞) with boundary T_{0}, and {μ_{t}}_{t≥0} is a family of mass measures of varifolds moving by mean curvature in the sense of Brakke. Let us write T_{t} for the time slice ∂(T ⌊(M × [t, ∞)) and μ_{T_{t}} for the mass measure of T_{t}. Then for all t ≥ 0, μ_{T_{t}} “sits atop” T_{t} in the sense that

μ_{t} ≥ μ_{T_{t}}

as Radon measures. Furthermore, {T_{t}}_{t≥0} is continuous in the weak topology on currents, and T_{t} is locally integral for each t.

The possible discrepancy between the two is an inherent flaw in the procedure, which arises due to the lower semicontinuity of mass with respect to weak convergence. In §9 we make the special assumption that T is unique with respect to the properties above. Employing a “restarting procedure”, we prove that {μ_{T_{t}}}_{t≥0} is itself a Brakke motion.

Such a current T is called a matching motion. Taking this as the definition of motion by mean curvature, we have gone some distance toward clarifying and sharpening Brakke’s definition.

In Chapter Two we prove that a codimension one surface (moving in R^{n}) can be perturbed slightly to one satisfying the uniqueness assumption. Using the results above,

ELLIPTIC REGULARIZATION 3

we find that the surface has unit density almost everywhere. This in turn is the crucial hypothesis for the application of Brakke’s theory of almost everywhere regularity. As a result we prove a generic almost-everywhere regularity theorem for the level-set flow.

§1 Varifolds and Currents

In order to follow the motion of a surface beyond the onset of singularities, we will replace the surface by its surface measure. Thus a motion by mean curvature will be a family {μ_{t}}_{t≥0} of Radon measures, such that for almost every t, μ_{t} possesses kdimensional tangent planes of positive integer multiplicity μ_{t}-almost everywhere. We take the liberty of calling these latter objects integer k-rectifiable Radon measures, and denote

IM_{k} = space of integer k-rectifiable Radon measures.

These are exactly the mass measures of integer rectifiable k-varifolds.

We will be needing the following spaces from geometric measure theory:

M(M) = space of Radon measures on M,
IV_{k}(M) = space of integer rectifiable k-varifolds on M,
V_{k}(M) = space of general k-varifolds on M,
I_{k}^{loc}(M) = space of locally integral k-currents on M,
D_{k}(M) = space of k-currents on M.

The basic references for these ideas are Federer [F], Almgren [A], and Allard [Al]; see also Morgan [M]. Our own source is mainly Simon [S]. We will devote this section to defining the above spaces and some other terms of geometric measure theory.

1.1 Radon Measures, Rectifiable Sets. Let (M, g) be a complete Riemannian manifold, assumed to be isometrically embedded in a high-dimensional Euclidean space. A Borel regular measure on M is an (outer) measure on M such that all Borel sets are measurable and every set A ⊆ M is contained in a Borel set of equal measure. A Radon measure on M is a locally finite Borel regular measure. The set of all Radon measure is denoted M(M).

For any Radon measure μ, define a linear functional μ on C_{c}^{0}(M, R) by

μ(ϕ) = ∫_{M} ϕ(x) dμ(x), ϕ ∈ C_{c}^{0}(M, R).

This formula sets up a one-to-one correspondence between the Radon measures on M and the set of continuous positive linear functionals on C_{c}^{0}(M, R). We endow M(M) with the topology such that μ_{i} → μ if and only if

μ_{i}(ϕ) → μ(ϕ) for every ϕ ∈ C_{c}^{0}(M, R).

For any set A ⊆ M, define the restriction of μ to A by

(μ⌊A)(B) = μ(A ∩ B) for B ⊆ M.

If ϕ is a locally μ-integrable function, define

(μ⌊ϕ)(B) = ∫_{B} ϕ dμ for B ⊆ M.

The k-dimensional density of μ at x is

Θ^{k}(μ, x) = lim_{r↓0} \frac{μ(B_{r}(x))}{ω_{k}r^{k}},

if the limit exists. Recalling that M is embedded in a Euclidean space, for any λ > 0 define the scaled Radon measure μ_{x,λ} by

μ_{x,λ}(A) = λ^{−k}μ(λA + x), A ⊆ M.

Let P be a k-plane in T_{x}M and a > 0. We say that P is the k-dimensional (approximate) tangent plane of μ with multiplicity a, written T_{x}μ = P, provided that

lim_{λ↓0} μ_{x,λ} = a H^{k}⌊P

where H^{k} is k-dimensional Hausdorff measure. The k-dimensional approximate tangent space of a set X ⊆ M at x is defined by

T_{x}X = T_{x}(H^{k}⌊X)

if this exists. We call X countably k-rectifiable (see Simon [S, 11.1]) provided

(i) X ⊆ C_{0} ∪ (⋃_{i≥1} C_{i}), where H^{k}(C_{0}) = 0 and each C_{i} is an embedded C^{1} k-

submanifold. We say X is locally k-rectifiable if in addition,

(ii) X has locally finite H^{k}-measure. If X is locally k-rectifiable and H^{k}-measurable, then T_{x}X exists H^{k}⌊X-a.e. (see [S, 11.6]).

6 TOM ILMANEN

1.2 k-Rectifiable Radon Measures. Let X be an H^{k}-measurable subset of M, and let θ : M → Z^{+} be locally H^{k}-integrable, with X = {θ > 0} H^{k}-a.e. Define μ(X, θ) to be the Radon measure

μ(X, θ) = H^{k}⌊θ.

Define μ to be a k-rectifiable Radon measure if either of the following equivalent conditions is satisfied:

(a) μ has k-dimensional tangent planes (of positive multiplicity) μ-a.e.

(b) μ = μ(X, θ) for X H^{k}-measurable and countably k-rectifiable, and θ locally

H^{k}-integrable. We write M_{k} for the set of k-rectifiable Radon measures. To see that (a) implies (b), apply [S, 11.8]. For the converse, see 1.3 below.

We call μ an integer k-rectifiable Radon measure if either of the following equivalent conditions is satisfied:

(c) μ has k-dimensional tangent planes of positive integer multiplicity μ-a.e.

(d) μ = μ(X, θ) for X H^{k}-measurable and locally H^{k}-rectifiable, and θ locally H^{k}-

integrable with values in Z^{+} ∪ {0}. We write IM_{k} for the set of all such μ. To see that (c) implies (d), observe that Θ^{k}(μ, x) ≥ 1 for μ-a.e. x ∈ M and apply [S, 11.8]. For the converse, see the following proposition.

1.3 Structure of μ(X, θ). Let μ be a Radon measure of the form μ(X, θ), where θ is locally H^{k}-integrable and X = {θ > 0} is countably k-rectifiable and H^{k}-measurable. Define

X (μ) = {x ∈ M : Θ^{k}(μ, x) exists and is positive}.

Then

(i) Θ^{k}(μ, x) exists and equals θ(x) for H^{k}-a.e. x ∈ M.
(ii) X = X (μ) H^{k}-a.e.
(iii) (a) For H^{k}-a.e. x ∈ M, either
T_{x}X exists, Θ^{k}(μ, x) > 0, and x ∈ X

ELLIPTIC REGULARIZATION 7

or

T_{x}X is empty, Θ^{k}(μ, x) = 0, and x ∉ X.

(b) Suppose X ⊆ C_{0} ∪ (⋃_{i≥1} C_{i}) where H^{k}(C_{0}) = 0 and each C_{i} is an embedded C^{1} k-submanifold. Then for H^{k}-a.e. x ∈ X, T_{x}μ = T_{x}C_{i} for every C_{i} ∋ x.

(c) If H^{k}⌊X is locally finite, then for H^{k}⌊X-a.e. x ∈ M, T_{x}μ = T_{x}X. (iv) (a) H^{k}⌊X has the same null-sets as μ.

(b) Θ^{k}(μ, x) = 0 for H^{k}-a.e. x ∈ spt μ \ X.

(c) μ(spt μ \ X) = 0. 1.4 Remarks.

(a) The point of (iv) is to emphasize that H^{k}(spt μ\X) need not equal zero. In fact, spt μ can be all of M, even though X is locally k-rectifiable.

(b) By (i) and (ii), θ and X are intrinsic to μ, up to sets of H^{k}-measure zero. We have for μ ∈ M_{k}(M),

μ = μ(X (μ), Θ^{k}(μ, ·)).

(c) Up to sets of H^{k}-measure zero, X (μ) is the smallest set supporting μ. 1.5 Proof of 1.3. We will use [S, 3.5, 4.7, 11.6].

1. We can assume X ⊆ C_{0} ∪ (∪_{i≥1}C_{i}), where H^{k}(C_{0}) = 0 and the C_{i} are disjoint embedded C^{1} submanifolds of M. Define the

X_{i} = X ∩ C_{i}, μ_{i} = μ⌊X_{i}.

According to Simon [S, 3.5], if μ is a Borel regular measure and A ⊆ R^{n} is μ-measurable with μ(A) < ∞, then Θ^{k}(μ⌊A, x) = 0 for H^{k}-a.e. x ∈ R^{n} \ A. Thus

Θ^{k}(μ, x) = Θ^{k}(μ_{i}, x)  for H^{k}-a.e. x ∈ C_{i},
Θ^{k}(μ, x) = 0  for H^{k}-a.e. x ∈ R^{n} \ X.

2. Thus for H^{k}-a.e. x ∈ C_{i},

Θ^{k}(μ, x) = Θ^{k}(μ_{i}, x) = \frac{dμ_{i}}{d(H^{k}⌊C_{i})} Θ^{k}(H^{k}⌊C_{i}, x) = θ(x)

8 TOM ILMANEN

by the Lebesgue differentiation theorem and the fact that μ_{i} = μ(C_{i}, θ⌊C_{i}). Together with step 1, this yields (i). Since X = {θ > 0} H^{k}-a.e., (i) implies (ii).

3. By steps 1 and 2, for H^{k}-a.e. x ∈ C_{i},

T_{x}μ = T_{x}μ_{i} = T_{x}C_{i}

with multiplicity θ(x), and T_{x}μ is empty for H^{k}-a.e. x ∈ M \ X. This yields (iii)(a), (b). If H^{k}⌊X is locally finite, then we get (iii)(c) by considering μ̃ = H^{k}⌊X.

4. Evidently μ ≪ H^{k}⌊X and conversely since θ > 0 on X, giving (iv)(a). Finally, (iv)(b) and (c) follow from (iii)(a) and the construction of μ, respectively. Q.E.D.

1.6 General Varifolds. A general k-varifold V is a Radon measure on G^{k}M, the Grassmann bundle of all k-planes tangent to M. We write

V (ψ) = ∫ ψ(x, S) dV (x, S)

where ψ ∈ C_{c}^{0}(G^{k}M, R) is a function on the total space of G^{k}M, and for each x ∈ M, S ranges over G_{x}^{k}M. We give the set V_{k}(M) of all general k-varifolds the topology of convergence of Radon measures.

Associated to V is a Radon measure μ_{V} on M, defined by μ_{V} = π_{#}(V), where π : G^{k}M → M is the projection map. Inversely, if μ is a k-rectifiable Radon measure, then by 1.3(iii)(c) T_{x}μ is a μ-measurable section of G^{k}M defined μ-a.e. Therefore it makes sense to define a varifold V_{μ} = V (μ) by the formula

V_{μ}(ψ) = ∫ ψ(x, S) dV_{μ}(x, S) = ∫ ψ(x, T_{x}μ) dμ(x) for ψ ∈ C_{c}^{0}(G^{k}M, R).

Varifolds of this form, for μ an integer k-rectifiable Radon measure, are called integer rectifiable k-varifolds. We write IV_{k}(M) for the set of integer rectifiable k-varifolds. The functions

V : IM_{k} → IV_{k}, μ : V_{k} → M

set up a one-to-one correspondence between IM_{k}(M) and IV_{k}(M). Note that μ is a continuous map but V is not.

ELLIPTIC REGULARIZATION 9

1.7 First Variation. (Simon [S, §16, §39]) If S is a k-dimensional subspace of a Euclidean space E, we will also use S to denote projection onto S, and

S · v = projection of the vector v onto S,
w · v = inner product of vectors w, v,
A : B = inner product of bilinear forms A and B.

Let V ∈ V_{k}(M) and μ = μ_{V}. Let Φ : M → M be C^{1}. The push-forward Φ_{#}(V) is defined by

Φ_{#}(V)(ψ) = ∫ ψ(x, S) d(Φ_{#}(V))(x, S) = ∫ ψ(x, S) |J_{S}Φ(x)| dV (x, S)

for ψ ∈ C_{c}^{0}(G^{k}M, R), where the Jacobian is given by

|J_{S}Φ(x)| = \sqrt{det[(dΦ(x)|S)^{T} ∘ dΦ(x)|S]}, x ∈ M, S ∈ G_{x}^{k}M.

The first variation δV of V is the linear functional on sections Y ∈ C_{c}^{1}(T M) given by

δV (Y) = \frac{d}{dt}|_{0}Φ_{#}^{t}(V) (U),

where spt Y ⊂⊂ U ⊂⊂ M and {Φ^{t}}_{t∈(−ε,ε)} is a family of diffeomorphisms supported in U with (∂/∂t)|_{0}Φ^{t} = Y, Φ^{0} = id. Letting ∇ denote covariant differentiation on M, we have the formula (see e.g. Simon [S, §9])

(1)  \frac{∂}{∂t}|_{0} |J_{S}Φ^{t}(x)| = S : ∇Y (x) = ∑_{i=1}^{k}⟨∇_{e_{i}}Y (x), e_{i}⟩

for any orthonormal basis {e_{1}, …, e_{k}} of S. (This is usually denoted in the literature by div_{V} Y.) Thus

δV (Y) = ∫_{V} S : ∇Y (x) dV (x, S).

For U ⊆ M open, define the total first variation by

|δV |(U) = sup {δV (Y) : Y ∈ C_{c}^{1}(T M|U), |Y | ≤ 1}.

If |δV | is a Radon measure, then δV is a T M-valued Radon measure with |δV | as total variation measure. In this case we can decompose δV as

δV = −μ⌊H⃗ + δV_{sing}

10 TOM ILMANEN

where H⃗ is a locally μ-integrable section of T M and δV_{sing} is supported on a set of μmeasure zero.

1.8 Abbreviations. When integrating over a k-rectifiable Radon measure μ, we will adopt the special notation

S = S(x) = T_{x}μ.

As observed above, S(x) is a μ-measurable section of G^{k}M defined μ-a.e. We will also write H⃗ for H⃗_{V_{μ}} in this context, and suppress the variable of integration whenever possible. Thus the first variation formula becomes

δV_{μ}(Y) = \frac{d}{dt}|_{0}Φ_{#}^{t}(V) (U) = ∫ S : ∇Y dμ = ∫ −H⃗ · Y dμ + δV_{sing}(Y).

1.9 Compactness for IM_{k}(M). (Allard) Let {μ_{i}}_{i≥1} ⊆ IM_{k}(M) with

sup_{i≥1} (μ_{i}(U) + |δV_{μ_{i}}|(U)) < ∞ for each U ⊂⊂ M.

Then there exists μ ∈ IM_{k}(M) and a subsequence {μ_{i_{j}}}_{j≥1} such that (i) Convergence of measure.

μ_{i_{j}} → μ as Radon measures on M.

(ii) Convergence of tangent planes.

V_{μ_{i_{j}}} → V_{μ} as Radon measures on G^{k}M.

(iii) Convergence of first variation.

δV_{μ_{i_{j}}} → δV_{μ} as T M-valued Radon measures.

(iv) Lower semicontinuity of total first variation.

|δV_{μ}| ≤ lim inf_{i_{j}} |δV_{μ_{i_{j}}} | as Radon measures.

The theorem is a restatement of Allard’s compactness theorem [Al, 6.4] in our notation. Note that (ii) implies (iii) by the first variation formula and (iii) implies (iv) because | · | is convex.

ELLIPTIC REGULARIZATION 11

1.10 Currents. Let Λ_{k}M denote the bundle of alternating k-vectors on M, and Λ^{k}M the corresponding bundle of k-forms. A k-current T ∈ D_{k}(M) is a continuous linear functional on the space C_{c}^{∞}(Λ^{k}M) of smooth k-forms of compact support. To any current T we associate the mass measure

μ_{T} (U) = sup {T (ϑ) : ϑ ∈ C_{c}^{∞}(Λ^{k}M|U), |ϑ| ≤ 1} , U ⊆ M open,

where |ϑ| is the norm of ϑ with respect to the metric on Λ_{x}^{k}M, normalized so that |η^{1} ∧ ⋯ ∧ η^{k}| = 1 for an orthonormal set {η^{1}, …, η^{k}} in T_{x}^{∗}M. The mass of T is

M[T] ≡ μ_{T} (M).

If μ_{T} is locally finite, then T is of locally finite mass. Such currents are also said to be representable by integration. In fact, by the Riesz representation theorem, μ_{T} is a Radon measure and

T (ϑ) = ∫ ⟨ϑ(x), ξ(x)⟩ dμ_{T} (x)

for some locally μ_{T} -integrable section ξ of Λ_{k}M. Thus in this case T is nothing more than a Λ_{k}M-valued Radon measure. (Here ⟨·, ·⟩ is the dual pairing.)

1.11 Locally Integer Rectifiable Currents. A locally integer rectifiable current is an integer k-rectifiable Radon measure endowed with a measurable choice of orientations for the tangent planes. Suppose we are given an integer k-rectifiable Radon measure μ = μ(X, θ). Let ξ be a μ-measurable section of Λ_{k}M|X such that

ξ(x) = e_{1} ∧ … ∧ e_{k}, for some orthonormal basis {e_{1}, …, e_{k}} of T_{x}μ

μ-almost everywhere. Then ξ(x) is called an orientation for T_{x}μ. Define the current T = τ(X, θ, ξ) by

T (ϑ) = ∫_{X}⟨ϑ(x), ξ(x)⟩ dμ(x) for ϑ ∈ C_{c}^{∞}(Λ^{k}M).

Such a current is called a locally integer rectifiable current, and we denote by R_{k}^{loc}(M) the space of all such currents.

Note that μ_{T} = μ, and the map

μ : R_{k}^{loc}(M) → IM_{k}(M)

is many-to-one and onto.

12 TOM ILMANEN

1.12 Boundaries. The boundary ∂T ∈ D_{k−1}(M) of a current T ∈ D_{k}(M) is defined by

∂T (ϑ) = T (dϑ), ϑ ∈ C_{c}^{∞}(Λ^{k}M).

A boundary is a current that can be expressed in this form, and a cycle is a current without boundary.

Note that if T ∈ R_{k}^{loc}(M), ∂T need not be locally rectifiable or even of locally finite mass. However, locally finite mass of the boundary implies local integer rectifiability, as follows.

1.13 Boundary Rectifiability Theorem. (Simon [S, 30.3]) Suppose T ∈ R_{k}^{loc}(M) and ∂T is of locally finite mass. Then ∂T ∈ R_{k−1}^{loc}(M).

A current T ∈ R_{k}^{loc}(M) with ∂T ∈ R_{k−1}^{loc}(M) is called a locally integral current. We write I_{k}^{loc}(M) for the set of locally integral k-currents.

If N^{k} is an oriented smooth submanifold of M, we can associate to N a locally integral current, denoted [N], by integrating a given k-form ϑ with respect to the oriented volume form of N.

1.14 Weak Topology for Currents. If {T_{i}}_{i≥1}, T ⊆ D_{k}(M), we say {T_{i}}_{i≥1} converges to T in the weak topology on D_{k}(M), written

T_{i} ⇀ T

provided that T_{i}(ϑ) → T (ϑ) for each ϑ ∈ C_{c}^{∞}(Λ^{k}M). Evidently, ∂ : D_{k}(M) → D_{k−1}(M) is continuous in the weak topology.

Suppose S, T ∈ I_{k}^{loc}(M). Fix U ⊆ M open. We define a pseudometric d_{U} by

d_{U}(S, T) = inf {M_{U}[P] + M_{U}[Q] : S − T = P + ∂Q, P ∈ R_{k}^{loc}(M), Q ∈ R_{k+1}^{loc}(M)}.

The topology generated by these pseudometrics as U ranges over all the open sets of M is called the flat metric topology. The two topologies are equivalent under the following circumstance.

1.15 Equivalent Topologies. (Simon [S, 31.2]) Let {T_{i}}_{i≥1}, T ⊆ I_{k}^{loc}(M) with

sup_{i≥1} (M_{U}[T_{i}] + M_{U}[∂T_{i}]) < ∞ for all U ⊂⊂ M.

ELLIPTIC REGULARIZATION 13

Then T_{i} ⇀ T if and only if d_{U}(T_{i}, T) → 0 for each U ⊂⊂ M.

Note: The “if” part is trivial and holds for all currents.

1.16 Weak Lower Semicontinuity of Mass. If T_{i} ⇀ T in D_{k}(M), then

dμ_{T} ≤ lim inf_{i→∞} dμ_{T_{i}}

as Radon measures.

In fact, the function μ : R_{k}^{loc}(M) → IM_{k}(M) (or μ : {currents of locally finite mass} → M(M)) is not weakly continuous. The inequality comes partly from the possibility that oppositely oriented portions may cancel in the limit. In contrast to the case of functions, the cancellation can occur even without large derivatives.

1.17 Compactness Theorem for I_{k}^{loc}(M). (Simon [S, 27.3]) Let {T_{i}}_{i≥1} ⊆ I_{k}^{loc}(M) satisfy

sup_{i≥1} (M_{U}[T_{i}] + M_{U}[∂T_{i}]) < ∞ for all U ⊂⊂ M.

Then there is T ∈ I_{k}^{loc}(M) and a subsequence {T_{i_{j}}}_{j≥1} such that T_{i_{j}} ⇀ T.

1.18 Slices and Boundaries. In this paper, we will often be dealing with a current P ∈ I_{k+1}^{loc}(M^{n} × R) with ∂P ∈ I_{k}^{loc}(M × {0}). Define

P_{z} = ∂(P ⌊(M × [z, ∞))).

Let us write (x, z) ∈ M × R, ¯∇ for the gradient on M × R, and D for the gradient on P, that is, the projection μ_{P} -a.e. of ¯∇ onto T_{(x,z)}P. Let

ζ : M × R → R

be the projection map. For each z ∈ R, define

X_{z} = X ∩ (M × \{z\}),  θ_{z} = θ|X_{z}  ξ_{z} = \{ \begin{matrix}ξ⌊\frac{Dζ}{|Dζ|} & if defined \\ anything & otherwise, \end{matrix}

where ξ ⌊ v signifies the contraction of the alternating (k + 1)-form ξ with the covector Dζ/|Dζ| to produce an alternating k-form. Then define

⟨P, ζ, z⟩ = τ(X_{z}, θ_{z}, ξ_{z}) ∈ I_{k}^{loc}(M),

14 TOM ILMANEN whenever X_{x}, θ_{z}, ξ_{z} are of the appropriate kind for such a definition. The following lemma says that this happens for a.e. z.

1.19 Slice Lemma. (Simon [S, 28.1, 28.5]) For L^{1}-a.e. z ∈ R, we have

(i) X_{z} is a locally k-rectifiable subset of M × {z}.

(ii) H^{k}⌊θ_{z} is a Radon measure.

(iii) For H^{k}-a.e. (x, z) ∈ X_{z}, both T_{(x,z)}X_{z} and T_{(x,z)}X exist and

T_{(x,z)}X = T_{(x,z)}X_{z} ⊕^{⊥} Span(Dζ)

(which implies Dζ ≠ 0) and ξ_{z} is a simple k-vector of unit length orienting T_{(x,z)}X_{z}.

(iv) ⟨P, ζ, z⟩ is properly defined, and P_{z} = ⟨P, ζ, z⟩ ∈ I_{k}^{loc}(M × {z}).

1.20 Co-Area Formula. (Simon [S, 28.1, 28.5]) For a μ_{P} -integrable function h = h(x, z) on X of compact support we have

∫_{P} h |Dζ| dμ_{P} = ∫_{−∞}^{∞} ∫_{P_{z}} h dμ_{P_{z}}dz

where we ignore the integrand ∫_{P_{z}} h dμ_{P_{z}} on the measure zero set of z for which P_{z} ∉ I_{k}^{loc}(M × {z}). In particular,

∫_{P ⌊(M×[z,∞))} h |Dζ| dμ_{P}

is an absolutely continuous function of z.

1.21 Slicing Level-Sets. As a special case of 1.19, let u : U → R be Lipschitz, where U ⊆ M is an oriented open set. Then graph(u) is locally n-rectifiable (see Simon [S, §11]). We form the n-current [u] ∈ I_{n}^{loc}(U × R) by

[u] = τ(graph(u), 1, ξ)

where ξ(x, z) ∈ (Λ_{k})_{x}(M × R) orients T_{(x,z)}graph(u) in agreement with the orientation induced from M, for H^{n}-a.e. (x, z) ∈ graph(u).

Then for a.e. z ∈ R, u^{−1}(z) is locally n-rectifiable and

⟨[u], ζ, z⟩ = τ(u^{−1}(z), 1, ξ_{z}) = ∂([u]⌊U × [z, ∞))⌊(U × R) = ∂[{u > z}]⌊U

ELLIPTIC REGULARIZATION 15

up to rigid motion, where ξ_{z} is as above, and [{u > z}] ∈ I_{n}^{loc}(U) is the current τ({u > z}, 1, e_{1} ∧ ⋯ ∧ e_{n}) for an oriented orthonormal basis {e_{1}, …, e_{n}} of T_{x}M.

1.22 Chart.

D_{k} ⊇ {Λ_{k}M-valued Radon measures} ⊇ R_{k}^{loc} ⊇  I_{k}^{loc}
μ↓^{onto}  ^{μ}↓^{onto}
M  ⊇ IM_{k}
μ↑^{onto} ∥
V_{k} ≡ {Radon measures on G^{k}M} ⊇ IV_{k} ⊇ {V with |δV | Radon}.

§2 Translative Functional

2.1 The Functional. Let (M^{n}, g) be a complete Riemannian manifold and fix k, 0 ≤ k ≤ n−1. Denote points in M ×R by w = (x, z). Fix ε > 0. We define the following functional for a current Q ∈ I_{k+1}^{loc}(M × R):

I^{ε}[Q] = \frac{1}{ε} ∫ e^{−z/ε}dμ_{Q}(x, z).

We shall see that a stationary point P^{ε} of I^{ε} is a translative soliton of the mean curvature flow with velocity −(1/ε)∂/∂z. That is, if permitted to move by mean curvature, P^{ε} will translate in the direction −∂/∂z with speed 1/ε.

2.2 Heuristics of Approximation by Elliptic Regularization. Fix a locally integral cycle T_{0} ∈ I_{k}^{loc}(M × {0}) of finite mass. We wish to approximate the motion of T_{0} by mean curvature. To do this, minimize I^{ε} over all currents Q ∈ I_{k+1}^{loc}(M × R) with ∂Q = T_{0}, to obtain a minimizer P^{ε}. For small ε, the fact that e^{−z/ε} dies off so quickly will cause P^{ε} to have great height, approximately C/ε. In fact, it will turn out that P^{ε} is nearly a cylinder, in any fixed finite region.

Let P_{z}^{ε} denote the slice of P^{ε} at the level z and denote the respective mean curvature vectors by H⃗_{P_{z}^{ε}}, H⃗_{P^{ε}}. Then

H⃗_{P_{z}^{ε}} ≈ H⃗_{P^{ε}}

for ε small, as shown in the picture. Since P^{ε} is a translative soliton,

Motion by H⃗_{P_{z}^{ε}} nearly produces motion (of P^{ε}) by −(1/ε)∂/∂z.

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 R^{n} × 0 P^{ε} H⃗_{P^{ε}}  T^{ε}T_{z}^{ε} P_{z}^{ε} H⃗_{P_{z}^{ε}}

Figure 1. P^{ε} and T^{ε}.

Define the rescaling function κ_{ε} : M × R → M × R by

κ_{ε}(x, z) = (x, εz) ≡ (x, t).

(Note that z and ε have units of length.) Define

T^{ε} = (κ_{ε})_{#}(P^{ε}), T_{t}^{ε} = slice of T^{ε} at height t.

Scaling, we find

Motion by H⃗_{T_{t}^{ε}} nearly produces motion (of T^{ε}) by −∂/∂t.

That is to say, T^{ε} is roughly the space-time track of T_{0} moving by mean curvature. We will pass to a subsequential limit T^{ε_{i}} ⇀ T and attempt to interpret T as the motion of T_{0} by mean curvature. Making this precise and exploring its consequences is the subject of this work.

2.3 Provenance. The idea of stretching out in spacetime so that H⃗_{P^{ε}} approximates H⃗_{P_{z}^{ε}} is due to Evans and Spruck. However, all their P ’s are smooth graphs. In fact, in the case H > 0, our T^{ε} is the graph of the function v^{ε} solving the equation [ES1, 7.16], namely

−(g − \frac{∇v^{ε} ⊗ ∇v^{ε}}{|∇v^{ε}|^{2}} ) : ∇^{2}v^{ε} = 1.

ELLIPTIC REGULARIZATION 17

This was the author’s starting point for this work.

The variational formulation of the soliton equation, for the case of homothetic solitons, was independently recorded by Angenent in [An4], based on the work of Huisken [H2].

2.4 Weighted First Variation. Let M be a manifold, and assume μ ∈ IM_{k}(M), Y ∈ C_{c}^{1}(T M), and {Φ^{s}}_{s∈(−δ,δ)} is a smooth family of diffeomorphisms with

\frac{∂}{∂s}|_{0} Φ^{s} = Y, Φ^{0} = id, Φ^{s}|(M \ U) = id for some U ⊂⊂ M.

Consider the functional

I[V_{μ}] = ∫_{M} f(x) dμ(x), f > 0 smooth.

The first variation of I is

\frac{∂}{∂s}|_{0} I[Φ_{#}^{s}(V_{μ}⌊U)] = \frac{∂}{∂s}|_{0} ∫ f(Φ^{s}(x))|J_{μ}Φ^{s}(x)| dμ(x)

= ∫ \frac{∂}{∂s}|_{0} f(Φ^{s}(x)) + f(x) \frac{∂}{∂s}|_{0} |J_{μ}Φ^{s}(x)| dμ

= ∫ ∇f · Y + fS : ∇Y dμ by 1.7(1)

= ∫ ∇f · S^{⊥} · Y + S : ∇(fY) dμ

= δV_{μ}(fY) + ∫ ∇f · S^{⊥} · Y dμ.

We obtain the weighted first variation formula

(1)  \frac{∂}{∂s}|_{0} I[Φ_{#}^{s}(V_{μ}⌊U)] = ∫ (−fH⃗ + ∇f · S^{⊥}) · Y dμ

if |δV | is absolutely continuous with respect to μ.

2.5 Notation. Let us write

∇ = ∇^{M}, ¯∇ = ¯∇^{M×R}, D = D^{P} , D̄ = D̄^{T}

for the gradient operators. Define ω = ∂/∂z.

We now specialize 2.4 to I^{ε}, replacing x by (x, z), ∇ by ¯∇ and

f(x, z) = \frac{1}{ε}e^{−z/ε},  ¯∇f = −\frac{1}{ε^{2}} e^{−z/ε}ω.

2.6 Euler-Lagrange Equation for I^{ε}. Assume that P^{ε} ∈ I_{k+1}^{loc}(M × R) is stationary with respect to I^{ε} on U ⊆ M × R, and μ^{ε} = μ_{P^{ε}}. Then for all Y ∈ C_{c}^{1}(T U), we have

0 = ∫ e^{−z/ε}(S : ¯∇Y − \frac{1}{ε}ω · Y) dμ^{ε}

= ∫ S : ¯∇(e^{−z/ε}Y) − \frac{1}{ε}e^{−z/ε}ω · S^{⊥} · Y dμ^{ε}

= ∫ S : ¯∇Y − \frac{1}{ε}ω · S^{⊥} · Y dμ^{ε} via Y ← e^{z/ε}Y

= ∫ (−H⃗ − \frac{1}{ε}S^{⊥} · ω) · Y dμ^{ε},

including absolute continuity of δV_{μ^{ε}} with respect to μ^{ε}. Therefore

(i) Equation: H⃗ + S^{⊥} · ω/ε = 0 μ^{ε}-a.e. asserting that P^{ε} is a translative soliton with velocity vector −ω/ε.

(ii) Bound on H⃗:  |H⃗| ≤ 1/ε  μ^{ε}-a.e.
(iii) Perpendicularity:  S^{⊥} · H⃗ = H⃗  μ^{ε}-a.e.

Now calculate, with ζ(x, z) ≡ z,

1 = | ¯∇ζ|^{2} = | ¯∇ζ · S^{⊥}|^{2} + | ¯∇ζ · S|^{2} = |ω · S^{⊥}|^{2} + |Dζ|^{2},

so by (i), we obtain the useful formula

(iv) Formula:  ε^{2}H^{2} + |Dζ|^{2} = 1  μ^{ε}-a.e.

§3 Existence of Approximators.

Let M be complete. Consider initial surfaces of the following kind.

3.1 Initial Surface. Assume that T_{0} ∈ I_{k}^{loc}(M × {0}) is a cycle of finite mass. With T_{0} as boundary, we will find stationary surfaces for I^{ε} by minimization.

Thereafter, only the stationary property will be used. The role of the minimizing property is under investigation.

ELLIPTIC REGULARIZATION 19

3.2 Existence of Stationary Surface with Boundary T_{0}. Assume M is complete and T_{0} satisfies 3.1. Then there exists P^{ε} ∈ I_{k+1}^{loc}(M × R) such that

(i) spt P^{ε} ⊆ M × [0, ∞), (ii) I^{ε}[P^{ε}] ≤ M[T_{0}], (iii) P^{ε} is stationary for I^{ε} in M × R \ spt T_{0}. We will prove this after 3.4. 3.3 Remarks. (a) As a matter of fact, it is easy to show that statements (ii) and

(iii) imply (i). On the other hand, if I^{ε}[P^{ε}] = ∞, then it is possible that spt P^{ε} ⊈ M × [0, ∞). For example, if T_{0} is stationary for the ordinary area integrand, take P = −T_{0} × [(−∞, 0]]. Another example is the z < 0 portion of any complete translating soliton, such as the grim reaper curve (see Angenent [An4] for an explanation) and the higher dimensional, rotationally symmetric translating solitons observed by Hamilton.

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Figure 2. Complete Translating Solitons.

(b) There are examples in which I^{ε}[P^{ε}] >> M[T_{0}]. For example, if we puncture a disk by a dense collection of very small disks, then what remains serves as P^{ε}.

(c) Another question is under what conditions (ii) becomes equality (for minimizers) as ε ↓ 0. If T_{0} is smooth, this holds.

3.4 Geometric Properties of I^{ε}. Note that I^{ε} is the area functional for the metric ḡ = e^{−2z/((k+1)ε)}(g ⊕ dz^{2}) where g ⊕ dz^{2} is the product metric on M × R. The curvature of ḡ is rather negative due to the convexity of the coefficient. When M = R^{n}, the sectional curvature of ḡ is

K(π) = \{ \begin{matrix}0 & for π containing ∂/∂z \\ −\frac{1}{n^{2}ε^{2}} e^{2z/((k+1)ε)} & for π parallel to R^{n} × \{0\}, \end{matrix}

where π is a 2-plane in R^{n} × R.

The metric ḡ is not complete, since ∫_{z_{0}}^{∞} e^{−z/((k+1)ε)}dz < ∞. In fact, the minimizers of I^{ε} can have a noncompact part trailing off the z ≫ 0 edge of M × R. Such minimizers approximate solutions of the mean curvature flow that exist for all time.

Proof of 3.2. The proof is standard. Let A = {Q ∈ I_{k+1}^{loc}(M × R) : ∂Q = T_{0}}. Now ∂(T_{0} × [[0, ∞)]) = T_{0} and

I^{ε}[T_{0} × [[0, ∞)]] = \frac{1}{ε} ∫_{T_{0}} ∫_{0}^{∞} e^{−z/ε} dz dμ_{T_{0}} = M[T_{0}] < ∞,

and therefore

inf_{Q∈A} I^{ε}[Q] ≤ M[T_{0}] < ∞.

Now let {P_{i}}_{i≥1} be a sequence of currents in A with

I^{ε}[P_{i}] ↓ inf_{Q∈A} I^{ε}[Q].

From this we obtain local mass estimates for {P_{i}}_{i≥1}. Hence by the compactness theorem 1.17, there exists P ∈ I_{k+1}^{loc}(M × R) with

P_{i} ⇀ P,  ∂P = T_{0},  μ_{P} ≤ lim inf μ_{P_{i}} as Radon measures,

and therefore

I^{ε}[P] ≤ lim inf_{i} I^{ε}[P_{i}] = inf_{Q∈A} I^{ε}[Q] ≤ M[T_{0}].

Thus P satisfies 3.2(ii) and minimizes I^{ε} in A . Therefore P is stationary for I^{ε} in M × R \ spt T_{0}. Also note that if μ_{P} (M × (−∞, 0)) > 0, the projection π^{+}(x, z) ≡ (x, max(z, 0)) reduces I^{ε}[P], so spt P^{ε} ⊆ M × [0, ∞). Q.E.D.

3.5 Compactness of Support. Assume the Ricci curvature of M is bounded below and that spt T_{0} is compact. Let P^{ε} be stationary with respect to I^{ε}, T^{ε} = (κ_{ε})_{#}(P^{ε}). Then for any τ > 0,

spt T^{ε} ∩ (M × [0, τ]) ⊆ K

ELLIPTIC REGULARIZATION 21

where K = K(spt T_{0}, τ) ⊆ M is a compact set independent of ε.

The idea is to prove the maximum principle for stationary varifolds touching smooth barriers (see Solomon-White [SW] or §10]), then to use the curvature hypothesis and the Bishop Comparison Theorem to construct barriers that resemble large, shrinking spheres disjoint from T_{0} (see [I1, §6]). Since we do not need this result, we will not prove it here.

3.6 Remarks. (a) Without the curvature hypothesis, a surface can rush off to infinity in finite time.

(b) Without the curvature hypothesis, a surface can blossom at infinity and rush inwards, on an initially empty manifold. While such surfaces are legitimate motions by mean curvature, they have very large mass (see 6.8) and cannot arise from minimizers P^{ε}.

The following proposition can be proven by similar computations.

3.7 Exterior Sphere Barrier. Let M = R^{n}. Suppose M = R^{n}, 1 ≤ k ≤ n − 1, and T_{0} ⊆ B_{R}(0). Then spt T^{ε} ⊆ {0 ≤ z ≤ (R^{2} − |x|^{2})/(2k)}.

This is consistent with the vanishing of a surface trapped within a shrinking sphere. It helps us to justify the scaling T^{ε} = (κ_{ε})_{#}(P^{ε}).

In the case k = 0, P^{ε} can consist of rays ±[{x} × [0, ∞)], or various grim reaper segments connecting points of T_{0}. These approximate points moving by mean curvature.

§4 H^{2} Estimate

If {T_{t}}_{t≥0} is a family of surfaces moving smoothly by mean curvature, then by inserting Y = H⃗ in the first variation formula 1.8, we obtain

\frac{d}{dt}M[T_{t}] = − ∫ H^{2} dμ

which yields the fundamental estimate for motion by mean curvature:

M[T_{t}] + ∫_{0}^{t} ∫_{T_{s}} H^{2} dμ_{T_{s}}ds = M[T_{0}].

We wish to derive an ε-analogue of this formula.

22 TOM ILMANEN

Let M be complete, let T_{0} satisfy 3.1, let P^{ε} ∈ I_{k+1}^{loc}(M × R) satisfy 3.2, and let μ^{ε} = μ_{P^{ε}}. Recall that ζ(x, z) = z, Dζ = S · ω. We will suppress the dependence on ε except in stating theorems.

4.1 Lemma. It is possible to insert Y = ξ(z)ω in 2.6 (first line), where ξ is Lipschitz. That is,

0 = ∫ e^{−z/ε}(ξ_{z}ω · S · ω − \frac{1}{ε}ξ) dμ^{ε}

for ξ : R → R Lipschitz with spt ξ ⊂⊂ [0, ∞).

Proof. First assume that ξ ∈ C_{c}^{1}((0, ∞), R). Fix a point x_{0} ∈ M. Let σ_{R} ∈ C_{c}^{1}(M, R^{+}) satisfy σ_{R} = 1 on B_{R}(x_{0}), σ_{R} = 0 off B_{2R}(x_{0}), 0 ≤ σ ≤ 1, and |∇σ_{R}| ≤ 2/R. Inserting Y = σ_{R}(x)ξ(z)ω in 2.6, we obtain

\begin{aligned}0 & = ∫ e^{−z/ε}(S : ¯∇Y − \frac{1}{ε}ω · Y) dμ \\ & = ∫ e^{−z/ε}(ξω · S · ¯∇σ_{R} + σ_{R}ξ_{z}ω · S · ω − \frac{1}{ε}ξσ_{R}) dμ. \end{aligned}

Now

|∫ e^{−z/ε}ξω · S · ¯∇σ_{R} dμ| ≤ \frac{2}{R} max |ξ|εI^{ε}[P]

so the first term disappears as we let R → ∞. For the rest of the integrand,

|e^{−z/ε}(σ_{R}ξ_{z}ω · S · ω − \frac{1}{ε}ξσ_{R})| ≤ (ε max |ξ_{z}| + max |ξ|)\frac{1}{ε}e^{−z/ε},

a fixed function in L^{1}(μ). Letting R → ∞, we obtain by the dominated convergence theorem,

(1)  0 = ∫ e^{−z/ε}(ξ_{z}ω · S · ω − \frac{1}{ε}ξ) dμ,

for smooth ξ.

2. Now let ξ : R → R be Lipschitz with spt ξ ⊂⊂ [0, ∞). Approximate ξ by ξ^{i} ∈ C_{c}^{1}((0, ∞)) such that

ξ^{i} ≤ ξ, ξ^{i} → ξ uniformly, ξ_{z}^{i} ⇀ ξ_{z} weakly-∗ in L^{∞}(R).

We can immediately pass to limits in the second term of (1). We use the co-area formula 1.20 to rewrite the first term:

∫ e^{−z/ε}ξ_{z}ω · S · ω dμ = ∫ e^{−z/ε}ξ_{z}|Dζ|^{2} dμ = ∫_{−∞}^{∞} ξ_{z} ∫_{P_{z}} e^{−z/ε}|Dζ|dμ_{P_{z}}dz.

ELLIPTIC REGULARIZATION 23

where P_{z} = ∂(P ⌊M × [z, ∞)). Now the function z ↦ ∫_{P_{z}} e^{−z/ε}|Dζ| dμ_{P_{z}} is in L^{1}(R), so we can pass to limits in the right-hand side as well. Q.E.D.

Replace ξ by ξe^{z/ε} in 4.1 to obtain

\begin{aligned}0 & = ∫ (\frac{1}{ε}ξ + ξ_{z}) ω · S · ω − \frac{1}{ε}ξ dμ \\ & = ∫ −\frac{1}{ε}ξω · S^{⊥} · ω + ξ_{z}ω · S · ω dμ \end{aligned}

By 2.6(i) we have S^{⊥} · ω = −εH⃗, and ω · S = Dζ, yielding 4.2 Lemma. Let ξ : R → R be Lipschitz with spt ξ ⊂⊂ [0, ∞). Then

∫ εξH^{2}dμ^{ε} = ∫ ξ_{z}|Dζ|^{2}dμ^{ε}.

We apply lemma 4.2 with ξ defined by

ξ(z) = \{ \begin{matrix}0 & on (−∞, z_{1}] \\ 1 & on [z_{1} + δ, z_{2}] \\ 0 & on [z_{2} + δ, ∞), \end{matrix}

and linearly interpolated between. We obtain (note that by the co-area formula, it makes no difference whether we use open or closed endpoints):

4.3 H^{2} Estimate, First Version. For 0 ≤ z_{1} ≤ z_{2} − δ,

\frac{1}{δ} ∫_{P^{ε}(z_{1},z_{1}+δ)} |Dζ|^{2} dμ^{ε} − \frac{1}{δ} ∫_{P^{ε}(z_{2},z_{2}+δ)} |Dζ|^{2} dμ^{ε} = ∫ εξH^{2} dμ^{ε}.

where P (a, b) denotes P ⌊(M × (a, b)).

The lemma implies that for any fixed δ, the function

f_{δ}(z) ≡ \frac{1}{δ} ∫_{P (z,z+δ)} |Dζ|^{2} dμ

is a decreasing function of z. Next we would like to estimate the quantity f_{δ}(0). Apply Lemma 4.1 with

ξ = ξ_{L,δ}(z) = \{ \begin{matrix}0 & on (−∞, 0] \\ 1 & at z = δ \\ 0 & on [δ + L, ∞), \end{matrix}

and linearly interpolated between. We have

ξ_{z}ω · S · ω = ξ_{z}|Dζ|^{2} = \{ \begin{matrix}\frac{1}{δ}|Dζ|^{2} & 0 ≤ z ≤ δ \\ −\frac{1}{L}|Dζ|^{2} & δ ≤ z ≤ δ + L. \end{matrix}

and 4.1 becomes

0 = \frac{1}{δ} ∫_{P (0,δ)} e^{−z/ε}|Dζ|^{2} dμ − \frac{1}{L} ∫_{P (δ,δ+L)} e^{−z/ε}|Dζ|^{2} dμ − ∫_{P} \frac{1}{ε}e^{−z/ε}ξ dμ.

The second term is bounded by (ε/L)I^{ε}[P], and therefore vanishes when L → ∞. Sending L → ∞, then δ → 0, we obtain

(1)  lim_{δ↓0} \frac{1}{δ} ∫_{P^{ε}(0,δ)} e^{−z/ε}|Dζ|^{2} dμ^{ε} ≤ I^{ε}[P^{ε}].

This leads to

4.4 Lemma. For any measurable set A ⊆ R,

\frac{1}{|A|} ∫_{P_{A}^{ε}} |Dζ|^{2} dμ^{ε} ≤ I^{ε}[P^{ε}],

where P_{A} = P ⌊M × A and |A| is the measure of A. In particular f_{δ}(0) ≤ I^{ε}[P^{ε}].

Proof of 4.4. The inequality holds for any open interval A by decomposing into many small intervals and using (1) together with the fact that f_{δ} is nonincreasing. Therefore it holds for all A. Q.E.D.

Define

f(z) = ∫_{P_{z}} |Dζ| dμ_{P_{z}}, z ∈ Z_{1},

where Z_{1} is the full measure subset of [0, ∞) on which P_{z} = ⟨P, ζ, z⟩ and the other good properties in the slice lemma 1.19 hold. By the co-area formula,

f_{δ}(z) = \frac{1}{δ} ∫_{z}^{z+δ} f(z′) dz′.

By the Lebesgue differentiation theorem and the decreasing property of f_{δ}, there is a full measure set Z ⊆ Z_{1} such that lim_{δ↓0} f_{δ}(z) = f(z) for z ∈ Z. Therefore we can pass to limits in 4.3, apply 4.4 and recall that I^{ε}[P] ≤ M[T_{0}] to obtain 4.5 H^{2} Estimate, Second Version. For z_{1}, z_{2} ∈ Z with 0 ≤ z_{1} ≤ z_{2}, we have

∫_{P_{z_{2}}^{ε}} |Dζ| dμ_{P_{z_{2}}^{ε}} + ∫_{P^{ε}(z_{1},z_{2})} εH^{2} dμ^{ε} = ∫_{P_{z_{1}}^{ε}} |Dζ| dμ_{P_{z_{1}}^{ε}} ≤ I^{ε}[P^{ε}] ≤ M[T_{0}].

§5 Space-Time Mass Bound

Let M be complete, and let T_{0}, P^{ε}, μ^{ε}, κ_{ε}, T^{ε} be as in 3.1, 3.2, 2.2. For A ⊆ R, define

P_{A} = P ⌊M × A,  T_{B} = T ⌊M × B,  B = εA.

We will obtain local mass estimates on M[T_{B}^{ε}] in order to pass to limits as ε_{i} ↓ 0. Let us suppress the dependence on ε except in stating theorems.

To motivate the estimate, suppose for a moment that T were the track in spacetime of a smooth surface {T_{t}}_{t≥0} evolving by mean curvature. Then by the co-area formula,

M[T (t, t + δ)] = ∫_{t}^{t+δ} ∫_{T_{s}} \frac{1}{|D̄τ|}dμ_{T_{y}}ds

where τ(x, t) = t, D̄ = D̄^{T}. If ds is a line element on T pointing in the direction of fastest time increase, we have

1/|D̄τ| = ds/dt = \sqrt{(dx/dt)^{2} + 1} = \sqrt{H^{2} + 1},

since a point x ∈ T_{t} is moving by ∂x/∂t = H⃗.

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Figure 3. Line Elements Recalling from §4 the estimate

M[T_{t}] + ∫_{0}^{t} ∫_{T_{s}} H^{2} dμ_{T_{s}}ds = M[T_{0}],

calculate

M[T (t, t + δ)] = ∫_{t}^{t+δ} ∫_{T_{s}} \sqrt{H^{2} + 1} dμ_{T_{s}}ds
≤ (∫_{t}^{t+δ} ∫_{T_{s}} H^{2} + 1 dμ_{T_{s}}ds)^{1/2} (∫_{t}^{t+δ} M[T_{s}] ds)^{1/2}
≤ (∫_{0}^{∞} ∫_{T_{s}} H^{2} dμ_{T_{s}}ds + δM[T_{0}])^{1/2} (δM[T_{0}])^{1/2}
≤ (1 + δ)^{1/2}δ^{1/2}M[T_{0}].

The above is the basic C^{1/2} estimate on the (k + 1)-volume crossed by T_{t}, combined with unit speed in the time direction.

Duly motivated, let us first estimate M[P_{A}]. Applying 2.6(iv), 4.4 and 4.5, we have with D = D^{P}, |A| = L^{1}(A),

5.1 Mass Bound for P^{ε}. Let M be complete, let T_{0}, P^{ε} satisfy 3.1, 3.2, and let A ⊆ R be measurable. Then

M[P_{A}^{ε}] = ∫_{P_{A}^{ε}} |Dζ|^{2} + ε^{2}H^{2} dμ ≤ (|A| + ε) I^{ε}[P^{ε}].

Now let π : M × R → M be the projection. Let us calculate M[π_{#}(T_{B})], where B = εA. We have

M[π_{#}(T_{B})] = M[π_{#}(P_{A})] ≤ ∫_{P_{A}} |J_{P} π| dμ

where |J_{P} π| is the determinant of the map dπ|T_{(x,t)}P : T_{(x,t)}P → T_{x}M. Now, dπ|T_{(x,t)}P has eigenvalues λ_{1} = λ_{2} = ⋯ λ_{k−1} = 1, λ_{k} = |J_{P} π|. From the diagram, evidently |J_{P} π|^{2} + |Dζ|^{2} = 1, but by 2.6(iv) we have ε^{2}H^{2} + |Dζ|^{2} = 1, so

|J_{P} π| = ε|H⃗|.

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Figure 4. Finding |J_{P} π|, |J_{P} κ_{ε}|.

Assuming that B is a finite union of open intervals with endpoints in the full measure good set Z of 4.5, calculate

M[π_{#}(T_{B})] ≤ ∫_{P_{A}} ε|H⃗| dμ^{ε}
≤ (εM[P_{A}])^{1/2} (∫_{P_{A}} εH^{2} dμ)^{1/2}
≤ (ε(ε + |A|)I^{ε}[P])^{1/2}I^{ε}[P]^{1/2} by 5.1, 4.5
= (ε^{2} + |B|)^{1/2}I^{ε}[P].

The result holds for any measurable B ⊆ R by approximation. Thus 5.2 C^{1/2} Estimate on (k+1)-Volume Crossed. For B ⊆ R measurable,

M[π_{#}(T_{B}^{ε})] ≤ ∫_{P_{A}^{ε}} |J_{P^{ε}}π| dμ^{ε} ≤ (ε^{2} + |B|)^{1/2}I^{ε}[P^{ε}].

In particular, for the flat metric distance we have the C^{1/2} + ε^{2} estimate

dist(π_{#}(T_{t}^{ε}), π_{#}(T_{t+δ}^{ε})) ≤ (ε^{2} + δ)^{1/2}I^{ε}[P^{ε}]

where T_{t}^{ε} = ∂(T [t, ∞)).

We now estimate M[T_{B}] by taking into account the time direction. By the Area Formula [S, 12.4],

M[T_{B}] = ∫_{P_{A}} |J_{P} κ_{ε}| dμ

where |J_{P} κ_{ε}| is the determinant of

dκ_{ε}|T_{(x,z)}P : T_{(x,z)}P → T_{(x,t)}T.

We calculate |J_{P} κ_{ε}| according to Figure 4,

|J_{P} κ_{ε}| = \sqrt{|J_{P} π|^{2} + ε^{2}|Dζ|^{2}} ≤ |J_{P} π| + ε|Dζ| ≤ |J_{P} π| + ε.

Thus

\begin{aligned}M[T_{B}] & ≤ ∫_{P_{A}} ε + |J_{P} π| dμ \\ & ≤ ε(\frac{|B|}{ε} + ε)I^{ε}[P] + (|B| + ε^{2})^{1/2}I^{ε}[P] by 5.1 and 5.2, \end{aligned}

and so we have proven,

5.3 Space-Time Mass Bound. Let M be complete. For T_{0}, P^{ε}, T^{ε} satisfying 3.1, 3.2, 2.2we have

M[T_{B}^{ε}] ≤ (|B| + ε^{2} + (|B| + ε^{2})^{1/2})I^{ε}[P^{ε}]

for B ⊆ R measurable.

§6 The Moving Varifolds of Brakke

K. Brakke introduced varifolds moving by mean curvature in his 1978 book [B]. While his method has some drawbacks, it is very general and has an impressive regularity theory.

Our definition involves Radon measures on M in place of the varifolds of [B]. For our purposes, nothing is lost by doing this, for V can be reconstructed from μ_{V} wherever it is k-rectifiable, and elsewhere V quickly loses mass independent of the choice of “abstract” tangent planes. We will check that our definition implies Brakke’s.

6.1 Motivation. Our staring point is the formula

(1)  \frac{d}{dt} ∫_{Γ_{t}} ϕ dH^{k} = ∫_{Γ_{t}} −ϕH^{2} + ∇ϕ · S^{⊥} · H⃗ dH^{k},

valid for a smooth family {Γ_{t}}_{t≥0} of submanifolds evolving by mean curvature, where ϕ : M → R is a smooth ambient test function. (Of course when M is smooth, we have S^{⊥} · H⃗ = H⃗). To derive this formula, apply the weighted first variation formula 2.4 with f = ϕ and ∂x/∂t = Y = H⃗ (smoothly extended to M). We now define a quantity B(μ, ϕ) to stand in for the right hand side of (1).

ELLIPTIC REGULARIZATION 29

6.2 Definition of B(μ, ϕ). Let M be complete, μ a Radon measure on M, and ϕ ∈ C_{c}^{2}(M, R^{+}).

Singular cases. In the following cases

(i) μ⌊{ϕ > 0} is not a k-rectifiable Radon measure, (ii) |δV |⌊{ϕ > 0} is not a Radon measure on {ϕ > 0}, where V = V_{μ}⌊{ϕ > 0}), (iii) |δV |⌊{ϕ > 0} is singular with respect to μ⌊{ϕ > 0}, (iv) ∫ ϕH^{2} dμ = ∞, where H⃗ = d(δV)/dμ⌊{ϕ > 0},

define B(μ, ϕ) = −∞.

Nonsingular case.

(v) Otherwise, define

B(μ, ϕ) = ∫ −ϕH^{2} + ∇ϕ · S^{⊥} · H⃗ dμ,

where ∇ = ∇^{M} and S = S(x) ≡ T_{x}μ for H^{k}-a.e. x ∈ {ϕ > 0}.

Note that the failure of each case renders the succeeding case meaningful. The upper derivate of f : R → R is defined by

D̄_{t_{0}}f(t) = lim sup_{t→t_{0}} \frac{f(t) − f(t_{0})}{t − t_{0}}.

6.3 Definition of Brakke Motion. Let M be complete and {μ_{t}}_{t≥0} a family of Radon measures on M. We call {μ_{t}}_{t≥0} a Brakke motion provided that for all t ≥ 0 and all ϕ ∈ C_{c}^{2}(M, R^{+}),

D̄_{t}μ_{t}(ϕ) ≤ B(μ_{t}, ϕ).

6.4 Discussion of Technical Points.

(a) It follows trivially from 6.2(i) and 7.2 that a Brakke motion satisfies μ_{t} ∈ M_{k}(M) for a.e. t ≥ 0, provided the mass stays locally bounded.

(b) Note that B(μ, ϕ) depends only on μ⌊{ϕ > 0}.

(c) Note that |δV |⌊{ϕ > 0} = |δV ⌊{ϕ > 0}|.

(d) It follows from lemma 6.6 below and Holder’s inequality that B(μ, ϕ) is finite in the nonsingular case 6.2(v).

30 TOM ILMANEN

(e) Brakke included the S^{⊥} before he had proven that for μ of locally finite total variation, S^{⊥} · H⃗ = 0 μ-a.e., see [B, §5.8]. We choose to retain it.

6.5 Advantages and Disadvantages of the Moving Varifolds of Brakke.

1. Inequality. It certainly seems preferable to have equality in 6.3. The inequality arises from the lower semicontinuity of ∫ H^{2}dμ in passing to a limit. In fact, there are moving varifolds where inequality is necessary. For example, if μ_{0} is stationary (i.e. δV_{μ_{0}} = 0), then equality in 6.3 would imply that μ_{t} = μ_{0} for all t ≥ 0. But a stationary, nonminimizing varifold really can begin moving, as shown in the figure.

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Figure 5. Stationary Cross Begins to Move.

2. Vanishing. The inequality permits μ_{t} to vanish abruptly, even gratuitously, or otherwise dwindle faster than expected. Part and parcel of the vanishing phenomenon is the question: in what sense does {μ_{t}}_{t≥0} assume the initial conditions?

In fact, certain varifolds must vanish abruptly. For example, Brakke [B, Fig. 9] gives the example of a spoon-shaped homothetic soliton, as in the figure. The varifold shrinks self-similarly toward the indicated point, then vanishes.

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Figure 6. Homothetic Spoon.

Conjecture. Sudden loss of mass never occurs for the motion of cycles (as distinct from surfaces with flanges, etc.).

ELLIPTIC REGULARIZATION 31

3. Nonuniqueness. Another characteristic of Brakke’s approach is nonuniqueness of the motion. The problem of gratuitous vanishing leads to much inessential nonuniqueness. However, nonuniqueness is a true property of motion by mean curvature, at least for nonsmooth initial data. The example in the figure above can be rotated to give nonuniqueness; Brakke also gives examples. Nonuniqueness occurs exactly when the corresponding level-set flow fattens, to be made precise in Chapter Two.

Conjecture. (Evans-Spruck; DeGiorgi) If Γ_{0} fattens under the level-set flow, then it fattens instantly. In particular, a smooth flow never fattens.

4. Multiple density. Brakke’s regularity theory requires the hypothesis:

For each t ≥ 0, μ_{t} has unit density μ_{t}-a.e.

However, Brakke’s existence theory does not provide this hypothesis.

We turn next to properties we will need. 6.6 Lemma. If ϕ ∈ C_{c}^{2}(M, R^{+}), then

\frac{|∇ϕ|^{2}}{ϕ} ≤ 2 max |∇^{2}ϕ| on \{ϕ > 0\}.

In particular, max |∇ϕ^{1/2}| < ∞.

Proof. (Y.-X. Zheng) Define f^{ε} = |∇ϕ|^{2}/(ϕ + ε). If ϕ ≢ 0, then f^{ε} will attain a maximum on {ϕ > 0}. At the maximum we have

0 = ∇f^{ε} = \frac{2∇^{2}ϕ · ∇ϕ}{ϕ + ε} − \frac{|∇ϕ|^{2}∇ϕ}{(ϕ + ε)^{2}}.

Since ∇ϕ ≠ 0 at the maximum of f^{ε}, we get

\frac{|∇ϕ|^{2}}{ϕ + ε} = 2 |∇^{2}ϕ · \frac{∇ϕ}{|∇ϕ|}| ≤ 2 max |∇^{2}ϕ|.

Letting ε ↓ 0, we obtain the desired inequality on {ϕ > 0}. Q.E.D.

6.7 Growth Bound. ([B, 3.4]) Let μ be a Radon measure and suppose ϕ ∈ C_{c}^{2}(M, R^{+}). Then

B(μ, ϕ) ≤ C_{2}(ϕ) μ({ϕ > 0}) < ∞

where C_{2}(ϕ) ≡ max |∇^{2}ϕ|/2.

32 TOM ILMANEN Proof. If B(μ, ϕ) > −∞, then

B(μ, ϕ) = ∫ −ϕH^{2} + ∇ϕ · S^{⊥} · H⃗ dμ

= ∫ −ϕ |H⃗ − \frac{1}{2} \frac{∇ϕ : S^{⊥}}{ϕ} |^{2} + \frac{1}{4} \frac{|∇ϕ : S^{⊥}|^{2}}{ϕ} dμ

≤ \frac{1}{4} ∫ \frac{|∇ϕ|^{2}}{ϕ} dμ

≤ (max |∇^{2}ϕ|/2) μ({ϕ > 0}) by 6.6.

Q.E.D.

6.8 Decreasing Property of Mass. If ∫_{0}^{t} M[μ_{s}] ds < ∞ then

M[μ_{t}] ≤ M[μ_{0}].

Proof. Fix x_{0} ∈ M. Let ϕ = ϕ_{R} ∈ C_{c}^{2}(M, R^{+}) satisfy ϕ = 1 on B_{R}(x_{0}), ϕ = 0 off B_{2R}(x_{0}), 0 ≤ ϕ ≤ 1, |∇ϕ| ≤ 2/R. Then by 6.3 and Cauchy’s inequality,

∫ ϕ^{2} dμ_{t} − ∫ ϕ^{2}dμ_{0} ≤ ∫_{0}^{t} B(μ_{s}, ϕ^{2}) ds ≤ ∫_{0}^{t} ∫ |∇ϕ|^{2} dμ_{s} ds ≤ \frac{4}{R^{2}} ∫_{0}^{t} M[μ_{s}] ds.

Letting R → ∞, we obtain the desired result. Q.E.D.

This shows, for example, that on those highly curved manifolds on which a surface can blossom at infinity, the surface must be pretty big at first.

6.9 Time-Varying Test Functions. Let {μ_{t}}_{t≥0} be a Brakke motion. Suppose ϕ ∈ C_{c}^{1}(M × [0, T], R^{+}). Then

D_{t}^{+}μ_{t}(ϕ(·, t)) ≤ B(μ_{t}, ϕ(·, t)) + ∫ \frac{∂}{∂t}ϕ(x, t) dμ_{t}(x).

Here D_{t_{0}}^{+}f(t) = lim sup_{t↓t_{0}}(f(t) − f(t_{0}))/(t − t_{0}). The proof is the same as [B, 3.5] except for the change of notation.

6.10 Brakke’s Original Definition. (adapted to M) Let V ∈ V_{k}(M) and ϕ ∈ C_{c}^{2}(M, R^{+}). Brakke [B, 3.2] defines δ(V, ϕ)(H⃗_{V}) as follows. If

(i) |δV |⌊ϕ is not a Radon measure, (ii) |δV |_{sing}⌊ϕ ≠ 0, where |δV |_{sing} is calculated with respect to μ_{V}, or (iii) ∫ ϕH_{V}^{2} dμ_{V} = ∞, where H⃗_{V} = d(δV ⌊{ϕ > 0})/dμ_{V}, then

δ(V, ϕ)(H⃗_{V}) = −∞.

(iv) Otherwise

δ(V, ϕ)(H⃗_{V}) = − ∫ ϕ(x)H^{2}(x) dμ_{V} (x) + ∫ ∇ϕ(x) · S^{⊥} · H⃗(x) dV (x, S).

A family {V_{t}}_{t≥0} ⊆ V_{k}(M) is said to be moving by mean curvature provided

D̄_{t}μ_{V_{t}}(ϕ) ≤ δ(V_{t}, ϕ)(H⃗_{V_{t}})

for all t ≥ 0 and ϕ ∈ C_{c}^{2}(M, R^{+}).

Let us see that our definition implies Brakke’s original definition. Given a Radon measure μ_{t} on M, define a varifold V_{t} ∈ V_{k}(M) as follows.

(i) On the largest open set U ⊆ R^{n} such that μ_{t}⌊U ∈ M_{k}(U), define V_{t}⌊(G^{k}M|U) = V (μ_{t}⌊U).

(ii) Elsewhere we extend V_{t} to be any varifold such that μ_{V_{t}} = μ_{t}.

If μ⌊{ϕ > 0} ∈ M_{k}({ϕ > 0}), then, using Holder’s inequality to handle 6.10(i), B(μ, ϕ) = δ(V, ϕ)(H⃗_{V}). Otherwise, B(μ, ϕ) = −∞. Therefore,

6.11 Implication Between the Definitions. If {μ_{t}}_{t≥0} satisfies 6.3, then any associated family {V_{t}}_{t≥0}, defined as above, satisfies 6.10.

The converse is not quite true, but this will not bother us.

§7 Convergence Lemma for Integral Brakke Motions

Like any other self-respecting elliptic or parabolic equation, Brakke’s motion has a compactness theorem. The proof is an adaptation of ideas in [B, Chap. 4]. We call a Brakke motion integral if μ_{t} ∈ IM_{k}(M) for a.e. t ≥ 0.

7.1 Compactness for Integral Brakke Motions. Let M be complete. Let {μ_{t}^{i}}_{t≥0}, i = 1, 2, … be a sequence of integral Brakke motions in M. Suppose

(i) sup_{i,t} μ_{t}^{i}(U) ≤ C_{1}(U) < ∞

34 TOM ILMANEN for each U ⊂⊂ M. Then

(ii) There is a subsequence {μ_{t}^{i_{j}}}_{t≥0}, j = 1, 2, … and an integral Brakke motion {μ_{t}}_{t≥0} such that

μ_{t}^{i_{j}} → μ_{t} as Radon measures for each t ≥ 0.

(iii) For a.e. t ≥ 0, there is a subsequence {i′_{j}}_{j≥1} of {i_{j}}_{j≥1} (depending on t) such

that

lim_{j→∞} V (μ_{t}^{i′_{j}}) = V (μ_{t}) as varifolds.

The proof appears in 7.6 after a few lemmas. Remarks. 1. Hypothesis (i) nearly always follows from mass bounds at the initial

time t = 0. See 6.8.

2. The theorem is equally valid for (real) k-rectifiable Brakke motions with a lower density bound

Θ^{k}(μ_{t}, x) ≥ θ_{0} > 0 μ_{t}-a.e. (for a.e. t).

7.2 Semidecreasing and Continuity Properties. Let {μ_{t}}_{t≥0} be a Brakke motion satisfying μ_{t}(U) ≤ C_{1}(U) for t ≥ 0. Then

(i) μ_{t} − C_{3}(ϕ)t is nonincreasing for ϕ ∈ C_{c}^{2}(M, R^{+}), where C_{3}(ϕ) = C_{2}(ϕ)C_{1}({ϕ > 0}).

(ii) The right and left limits always exist and satisfy

lim_{s↑t} μ_{s}(ϕ) ≥ μ_{t}(ϕ) ≥ lim_{s↓t} μ_{s}(ϕ).

(iii) There is a co-countable set B_{2} ⊆ [0, ∞) with μ_{t} continuous at all t ∈ B_{2}. (iv) There is a full measure set B_{3} ⊆ [0, ∞) with D̄_{t}μ_{t}(ϕ) > −∞ for all ϕ ∈

C_{c}^{2}(M, R^{+}) and all t ∈ B_{3}.

(v) μ_{t} ∈ M_{k}(M) for a.e. t ≥ 0. Proof. (i) By 6.3 and 6.7, we have

D̄_{t}μ_{t}(ϕ) ≤ B(μ_{t}, ϕ) ≤ C_{2}(ϕ)C_{1}({ϕ > 0}).

(ii) is clear from (i).

ELLIPTIC REGULARIZATION 35

(iii) Let Ψ be a countable dense subset of C_{c}^{2}(M, R^{+}). By (i), for each ψ ∈ Ψ there is a co-countable set B_{ψ} ⊆ [0, ∞) such that μ_{t}(ψ) is continuous on B_{ψ}. Define B_{2} = ⋂_{ψ∈Ψ} B_{ψ}, so that μ_{t}(ψ) is continuous at each t ∈ B_{2} for each ψ ∈ Ψ. The result follows by uniform approximation.

(iv) For each ψ ∈ Ψ, there is a full measure set B̃_{ψ} such that D̄_{t}μ_{t}(ϕ) > −∞ on B̃_{ψ}. Define B_{3} = ⋂_{ψ∈Ψ} B̃_{ψ}. Let ϕ ∈ C_{c}^{2}(M, R^{+}) and choose ψ ∈ Ψ, ψ ≥ ϕ. For t ∈ B_{3},

−∞ < D̄_{t}μ_{t}(ψ) ≤ D̄_{t}μ_{t}(ϕ) + D̄_{t}μ_{t}(ψ − ϕ) ≤ D̄_{t}μ_{t}(ϕ) + C_{3}(ψ − ϕ)

by (1), so D̄_{t}μ_{t}(ϕ) > −∞ for t ∈ B_{3}.

(v) Therefore by definition of B(μ_{t}, ϕ), we have μ_{t}⌊{ϕ > 0} ∈ M_{k}({ϕ > 0}) for all t ∈ B_{3}. Q.E.D.

7.3 Upper Semicontinuity of B(μ, ϕ). ([B], 4.28]) Let ϕ ∈ C_{c}^{2}(M, R^{+}). Let {μ_{i}}_{i≥1} ⊆ IM_{k}(M) be a sequence satisfying

(i)  B(μ_{i}, ϕ) ≥ −C_{4}, μ_{i}({ϕ > 0}) ≤ C_{4}  for all i ≥ 1.

Then there is a subsequence {μ_{i_{j}}}_{j≥1} and a Radon measure μ ∈ IM_{k}({ϕ > 0}) such that

(ii)

μ_{i_{j}}⌊{ϕ > 0} → μ  as Radon measures on {ϕ > 0}
V (μ_{i_{j}}⌊{ϕ > 0}) → V (μ)  as varifolds over {ϕ > 0}.

Furthermore, for any convergent sequence μ_{i} → μ of Radon measures on {ϕ > 0} with sup_{i≥1} μ_{i}({ϕ > 0}) < ∞, we have

(iii)  lim sup_{i} B(μ_{i}, ϕ) ≤ B(μ, ϕ).

To prove this, we need the following elementary lemma.

7.4 L^{2} Approximation Lemma. Let μ be k-rectifiable. Then for any vector field Z ∈ L^{2}(μ, T M) and any ε > 0, there is a vector field Y ∈ C_{c}^{1}(T M) such that

∥Z − Y ∥_{L^{2}(μ,T M)} ≤ ε.

Proof. We can assume μ = μ(X, θ) where X ⊆ ⋃_{i=1}^{∞} C_{i} and the C_{i} are disjoint, precompact C^{1} submanifolds of M. In L^{2}(μ, T M) we have Z = ∑_{i} Z|C_{i}. Each function 36 TOM ILMANEN

Z|C_{i} can be well approximated in L^{2}(H^{k}⌊C_{i}⌊θ, T M) by a function Z_{i} with spt Z_{i} ⊂⊂ C_{i}. Then any finite assemblage Z_{1}, …, Z_{p} can be approximated in L^{2}(H^{k}⌊C_{i}⌊θ, T M) by functions Y_{1}, …, Y_{p} ∈ C_{c}^{1}(T M) with disjoint supports. Then Y_{1} + ⋯ + Y_{p} approximates Z in L^{2}(μ, T M). Q.E.D.

Proof of 7.3. 1. Let μ_{i}, i = 1, 2, … satisfy inequalities (i). Then V_{μ_{i}}, H⃗_{μ_{i}} make sense and using Cauchy’s inequality,

−C_{4} ≤ ∫ −ϕH^{2} + ∇ϕ · S^{⊥} · H⃗ dμ_{i} ≤ ∫ −\frac{1}{2}ϕH^{2} + \frac{1}{2} \frac{|∇ϕ|^{2}}{ϕ} dμ_{i},

from which we obtain

(1)  ∫ ϕH^{2} dμ_{i} ≤ 2C_{4} + 4C_{4}C_{2}(ϕ) = C(C_{4}, ϕ).

and by Cauchy-Schwarz, for each U ⊂⊂ {ϕ > 0},

(2)  |δV_{μ_{i}}|(U) ≤ C(C_{4}, ϕ, U).

Since also μ_{i}({ϕ > 0}) ≤ C_{4}, Allard’s compactness theorem 1.9 implies that there is a subsequence {μ_{i_{j}}}_{j≥1} and a Radon measure μ ∈ IM_{k}({ϕ > 0}) such that

μ_{i_{j}}⌊{ϕ > 0} → μ  as Radon measures on {ϕ > 0},
V (μ_{i_{j}}⌊{ϕ > 0}) → V (μ) as varifolds over {ϕ > 0},

proving (ii).

2. For (iii), let μ_{i} → μ as Radon measures on {ϕ > 0} with sup_{i≥1} μ_{i}({ϕ > 0}) < ∞. If lim sup_{i} B(μ_{i}, ϕ) = −∞, we are done. Otherwise assume, by taking a subsequence, that lim_{i→∞} B(μ_{i}, ϕ) exists and is finite. Therefore hypothesis (i) holds for {μ_{i}}_{i≥1}. By (ii), we can replace {μ_{i}}_{i≥1} by a subsequence and assume (after relabelling) that

(3)  V (μ_{i}⌊{ϕ > 0}) → V (μ⌊{ϕ > 0})

as varifolds over {ϕ > 0}.

3. We cannot prove (iii) directly for ϕ. We will prove it first for ψ ∈ C_{c}^{2}({ϕ > 0}, R^{+}). Note by (1) and (2) that both ∫ −ψH^{2} dμ_{i} and ∫ ∇ψ · S^{⊥} · H⃗ dμ_{i} are bounded independent of i.

4. For the first of these terms, by the rectifiability of μ and the L^{2} Approximation Lemma we have

(∫ ψH^{2} dμ)^{1/2} = sup {∫ ψ^{1/2}H⃗ · Y dμ : ∥Y ∥_{L^{2}(μ,T M)} = 1, Y ∈ C_{c}^{1}({ϕ > 0}, T M)} .

Recalling that ψ^{1/2} ∈ C^{1} by 6.6, calculate

∫ ψ^{1/2}H⃗ · Y dμ = −δV_{μ}(ψ^{1/2}Y)

= lim_{i→∞} −δV_{μ_{i}}(ψ^{1/2}Y) by (3)

= lim_{i→∞} ∫ ψ^{1/2}H⃗_{i} · Y dμ_{i}

≤ lim inf_{i→∞} (∫ ψH_{i}^{2} dμ_{i})^{1/2} (∫ |Y |^{2}dμ_{i})^{1/2}

from which it follows that

(4)  ∫ ψH^{2} dμ ≤ lim inf_{i→∞} ∫ ψH_{i}^{2} dμ_{i}.

5. For the second term, note that ∇ψ(x) · S^{⊥}(x) is a μ-integrable vector field in L^{∞}(μ, T M). Since μ is k-rectifiable, by the L^{2} Approximation Lemma we can choose Y ∈ C_{c}^{1}({ϕ > 0}, R^{n}) with

∥Y − ∇ψ · S^{⊥}∥_{L^{2}(μ,T M)} ≤ δ.

Then

∫ ∇ψ · S^{⊥} · H⃗ dμ − ∫ ∇ψ · S_{i}^{⊥} · H⃗_{i} dμ_{i} = ∫ (∇ψ · S^{⊥} − Y) · H⃗ dμ  (A)
− δV_{μ}(Y) + δV_{μ_{i}}(Y)  (B)
+ ∫ (Y − ∇ψ · S_{i}^{⊥}) · H⃗_{i} dμ_{i}.  (C)

Now lim_{i→∞} B = 0 by (3) and

|A|^{2} ≤ δ^{2} ∫_{\{ψ>0\}} H^{2}dμ ≤ δ^{2} sup_{\{ψ>0\}} (\frac{1}{ϕ}) ∫ ϕH^{2}dμ ≤ C(ϕ, ψ)δ^{2},

38 TOM ILMANEN and again by (3),

lim_{i→∞} |C|^{2} ≤ sup_{\{ψ>0\}} (\frac{1}{ϕ}) sup_{i>0} ∫ ϕH_{i}^{2} dμ_{i} · lim_{i→∞} ∫_{\{ψ>0\}} |Y (x) − ∇ψ(x)S^{⊥}|^{2}dV_{μ_{i}}(x, S)

≤ C(ϕ, ψ) ∫_{{ψ>0}} |Y (x) − S^{⊥} · ∇ψ(x)|^{2}dV_{μ}(x, S)

≤ C(ϕ, ψ)δ^{2}.

We pass δ → 0 in (A), (B), (C) to obtain

lim_{i→∞} ∫ ∇ψ · S_{i}^{⊥} · H⃗_{i} dμ_{i} = ∫ ∇ψ · S^{⊥} · H⃗ dμ.

Combining with (4), we get

(5)  lim sup_{i→∞} B(μ_{i}, ψ) ≤ B(μ, ψ)  for ψ ∈ C_{c}^{2}({ϕ > 0}, T M).

6. Now choose a sequence {ψ_{j}}_{j≥1} in C_{c}^{2}({ϕ > 0}, R^{+}) such that ψ_{j} ≤ ϕ and ψ_{j} → ϕ in C^{2}. Then

lim_{j→∞} ∫ ψ_{j}H^{2} dμ = ∫ ϕH^{2} dμ

by the dominated convergence theorem. Also

|∫ ∇ψ_{j} · S^{⊥} · H⃗ dμ − ∫ ∇ϕ · S^{⊥} · H⃗ dμ|^{2} ≤ ∫ \frac{|∇ψ_{j} − ∇ϕ|^{2}}{|ψ_{j} − ϕ|} dμ ∫ |ψ_{j} − ϕ|H^{2}dμ.

Each of the terms on the right goes to zero as j → ∞ since ∫ ϕH^{2}dμ < ∞ and ψ_{j} → ϕ in C^{2} (see 6.6). Combining these,

(6) lim_{j→∞} B(μ, ψ_{j}) = B(μ, ϕ).

Then by (5) and 6.7,

lim sup_{i→∞} B(μ_{i}, ϕ) ≤ lim sup_{i→∞} B(μ_{i}, ψ_{j}) + sup_{i>0} B(μ_{i}, ϕ − ψ_{j})

≤ B(μ, ψ_{j}) + 2∥ϕ − ψ_{j}∥_{C^{2}} sup_{i>0} μ_{i}({ϕ > 0}).

Then by (6), sending j → ∞,

lim sup_{i→∞} B(μ_{i}, ϕ) ≤ B(μ, ϕ)

ELLIPTIC REGULARIZATION 39

as desired. Q.E.D.

7.5 Measurability. Let {μ_{t}}_{t≥0} be a Brakke motion satisfying μ_{t}(U) ≤ C_{1}(U) for U ⊂⊂ M. Then

(i) t ↦ D̄_{t}μ_{t}(ϕ) is measurable.

(ii) t ↦ B(μ_{t}, ϕ) is upper semicontinuous on the set B_{2} of points at which {μ_{t}}_{t≥0}

is continuous. Proof. (i) follows from 7.2(i). For (ii), if t_{i} → t and B(μ_{t_{i}}, ϕ) ≥ α, then by the

upper semicontinuity result 7.3, there is a Radon measure μ on M and a subsequence with

μ_{t_{i_{j}}} → μ, B(μ, ϕ) ≥ α.

If t ∈ B_{2}, then μ = μ_{t}. Q.E.D.

7.6 Proof of 7.1. Idea of proof: We will first establish subsequential convergence of the measure μ_{t}^{i} on a countable dense set of times. Convergence for all times follows by the semidecreasing property in 7.2. The upper semicontinuity 7.3 allows Brakke’s inequality 6.3 to pass to limits. The estimate ∫∫ H^{2} dμ_{t} dt < ∞ and Allard’s compactness theorem 1.9 proves integer rectifiability for a.e. t.

1. Let B_{1} be a countable dense set of times. For each t ∈ B_{1}, the mass bound 7.1(i) and the compactness of Radon measures allows us to choose a subsequence {μ_{t}^{i_{j}}}_{j≥1} and a Radon measure μ_{t} on M such that μ_{t}^{i_{j}} → μ_{t} as Radon measures. By choosing successive subsequences and diagonalizing, we can find a single subsequence, also to be called {μ_{t}^{i_{j}}}_{j≥1}, such that

μ_{t}^{i_{j}} → μ_{t} for all t ∈ B_{1}.

For any ϕ ∈ C_{c}^{2}(M, R^{+}), by 7.2(i) we know that μ_{t}^{i_{j}}(ϕ) − C_{3}(ϕ)t is nonincreasing in t. Passing to limits as i_{j} → ∞ we find

μ_{t}(ϕ) − C_{3}(ϕ)t is nonincreasing for t ∈ B_{1}.

2. For t ∈ [0, ∞) \ B_{1}, let {μ_{t}^{i′_{k}}}_{k≥1} be any convergent subsequence of {μ_{t}^{i_{j}}}_{j≥1}, depending on t. Define the Radon measure μ_{t} by

μ_{t} = lim_{k→∞} μ_{t}^{i′_{k}}.

So μ_{t} is defined for all t ∈ [0, ∞). Applying step 1 to B_{1} ∪ {t} separately for each t ∈ [0, ∞) \ B_{1}, we find

μ_{t}(ϕ) − C_{3}(ϕ)t is nonincreasing for each ϕ ∈ C_{c}^{2}(M, R^{+}).

By 7.2(iii), there is a co-countable set B_{2} such that μ_{t} is continuous at all t ∈ B_{2}. For such t, μ_{t} is uniquely determined, independent of the subsequence {i′_{k}}_{k≥1} chosen for that t, and hence the full sequence converges:

μ_{t}^{i_{j}} → μ_{t} for t ∈ B_{2}.

Because [0, ∞) \ B_{2} is countable, we can choose further subsequences and diagonalize again to obtain a single subsequence, also to be called {i_{j}}_{j≥1}, such that lim_{i_{j}→∞} μ_{t}^{i_{j}} exists for all t ≥ 0. For t ∈ [0, ∞) \ B_{2}, replace the old values of μ_{t} by these new ones. Thus we have constructed a family {μ_{t}}_{t≥0} of Radon measures such that

(1) μ_{t}^{i_{j}} → μ_{t} for all t ≥ 0,

μ_{t}(ϕ) − C_{3}(ϕ)t is nonincreasing for all ϕ ∈ C_{c}^{2}(M, R^{+}).

3. Let us prove that {μ_{t}}_{t≥0} is an integral Brakke motion. Fix ϕ ∈ C_{c}^{2}(M, R^{+}) and t_{0} ≥ 0. We aim to show that D̄_{t_{0}}μ_{t}(ϕ) ≤ B(μ_{t_{0}}, ϕ). Now D̄_{t_{0}}μ_{t}(ϕ) = max(D_{t_{0}}^{−}μ_{t}(ϕ), D_{t_{0}}^{+}μ_{t}(ϕ)), where D^{−} and D^{+} are the left and right upper derivates, respectively. We will prove that D_{t_{0}}^{+}μ_{t}(ϕ)) ≤ B(μ_{t_{0}}, ϕ); the other case is similar.

Without loss of generality, assume D_{t_{0}}^{+}μ_{t}(ϕ) > −∞. Then there exist t_{q} ↓ t_{0}, ε_{q} ↓ 0 with

(2)  D_{t_{0}}^{+}μ_{t}(ϕ) − ε_{q} ≤ \frac{μ_{t_{q}}(ϕ) − μ_{t_{0}}(ϕ)}{t_{q} − t_{0}}.

By the convergence μ_{t}^{i_{j}} → μ_{t}, there is for each q a number r_{q}, selected from the sequence {i_{j}}_{j≥1}, such that

D_{t_{0}}^{+}μ_{t}(ϕ) − 2ε_{q} ≤ \frac{μ_{t_{q}}^{r_{q}}(ϕ) − μ_{t_{0}}^{r_{q}}(ϕ)}{t_{q} − t_{0}} ≤ \frac{1}{t_{q} − t_{0}} ∫_{t_{0}}^{t_{q}} D̄_{t}μ_{t}^{r_{q}}(ϕ) dt.

Note that D̄_{t}μ_{t}^{r_{q}}(ϕ) is measurable and bounded above by C_{3}(ϕ). We can assume that r_{q} → ∞ as q → ∞. It follows that there is at least one point s_{q} ∈ [t_{0}, t_{q}] such that μ_{s_{q}}^{r_{q}} ∈ IM_{k}({ϕ > 0}) and

D_{t_{0}}^{+}μ_{t}(ϕ) − 2ε_{q} ≤ D̄_{t}μ_{s_{q}}^{r_{q}}(ϕ) ≤ B(μ_{s_{q}}^{r_{q}}, ϕ).

By the mass bound 7.1(i) and the upper semicontinuity lemma 7.3, there is a subsequence {μ_{s′_{q}}^{r′_{q}}}_{q≥1} and a Radon measure μ ∈ IM_{k}({ϕ > 0}) such that

(3)  r′_{q} → ∞,  s′_{q} ↓ t_{0},  μ_{s′_{q}}^{r′_{q}} → μ as Radon measures

and

(4)  D_{t_{0}}^{+}μ_{t}(ϕ) ≤ lim sup_{q→∞} B(μ_{s′_{q}}^{r′_{q}}, ϕ) ≤ B(μ, ϕ).

4. We must check that

μ = μ_{t_{0}}⌊{ϕ > 0}.

Let ψ ∈ C_{c}^{2}({ϕ > 0}, R^{+}). Assume by scaling that ψ ≤ ϕ. Then

−∞ < D_{t_{0}}^{+}μ_{t}(ϕ) ≤ D_{t_{0}}^{+}μ_{t}(ψ) + D_{t_{0}}^{+}μ_{t}(ϕ − ψ) ≤ D_{t_{0}}^{+}μ_{t}(ψ) + C_{3}(ϕ − ψ)

by 7.2(i). Thus D_{t_{0}}^{+}μ_{t}(ψ) > −∞, and using 7.2(i) to control the times not covered by the lim sup,

(5)  μ_{t_{0}}(ψ) ≤ lim_{t↓t_{0}} μ_{t}(ψ).

Next we will show that μ_{t_{0}}(ψ) = μ(ψ). Fix t > t_{0}, then for q large enough we have t_{0} < s′_{q} < t. Since μ_{t}^{i}(ψ) − C_{3}(ψ)t is nonincreasing in t,

μ_{t_{0}}^{r′_{q}}(ψ) + C_{3}(ψ)(s′_{q} − t_{0}) ≥ μ_{s′_{q}}^{r′_{q}}(ψ) ≥ μ_{t}^{r′_{q}}(ψ) + C_{3}(ψ)(s′_{q} − t).

Sending r′_{q} → ∞ and s′_{q} ↓ t_{0}, we obtain by (3) and (1),

μ_{t_{0}}(ψ) ≥ μ(ψ) ≥ μ_{t}(ψ) + C_{3}(ψ)(t_{0} − t)

for any t > t_{0}. Sending t ↓ t_{0}, we obtain by (5)

μ_{t_{0}}(ψ) ≥ μ(ψ) ≥ lim_{t↓t_{0}} μ_{t}(ψ) ≥ μ_{t_{0}}(ψ).

Thus μ_{t_{0}}(ψ) = μ(ψ). This was true for arbitrary ψ ∈ C_{c}^{2}({ϕ > 0}, R^{+}), so by (4) and 6.4(b),

D_{t_{0}}^{+}μ_{t}(ϕ) ≤ B(μ, ϕ) = B(μ_{t}, ϕ).

42 TOM ILMANEN

The case of D_{t_{0}}^{−}μ_{t}(ϕ) is similar, and so D̄_{t_{0}}μ_{t}(ϕ) ≤ B(μ_{t}, ϕ). Together with (1), we have proven 7.1(ii).

5. Let us prove (iii). Let Ψ be a countable subset of C_{c}^{2}(M, R^{+}) such that the open sets {ϕ > 0} for ϕ ∈ Ψ together cover M. Fix T < ∞. For each ϕ ∈ Ψ, by the mass bound 7.1(i)

−C ≤ μ_{T}^{i_{j}}(ϕ) − μ_{0}^{i_{j}}(ϕ) ≤ ∫_{0}^{T} D̄_{t}μ_{t}^{i_{j}}(ϕ) dt ≤ ∫_{0}^{T} B(μ_{t}^{i_{j}}, ϕ) dt

where C = C(ϕ, T). Then

−C ≤ lim sup_{j→∞} ∫_{0}^{T} B(μ_{t}^{i_{j}}, ϕ) dt ≤ ∫_{0}^{T} lim sup_{j→∞} B(μ_{t}^{i_{j}}, ϕ) dt.

Thus for a.e. t ≥ 0,

lim sup_{j→∞} B(μ_{t}^{i_{j}}, ϕ) > −∞.

In fact, for a.e. t ≥ 0, the above statement holds for every ϕ ∈ Ψ. Then by 7.1(i), (ii) and the B-compactness lemma 7.3, there is a subsequence {i′_{j}}_{j≥1} of (i_{j}}_{j≥1}, depending on t, such that

μ_{t}^{i′_{j}} → μ_{t} ∈ IM_{k}(M), V (μ_{t}^{i′_{j}}) → V (μ_{t}).

We have proven 7.1(iii). Q.E.D.

§8 Enhanced Motion

Given an initial surface T_{0}, we will use the estimates in §4-§5 and the Brakke convergence theorem of §7 to produce a pair

(T, {μ_{t}}_{t≥0})

that we call “enhanced motion.” Here T is a locally integral current with boundary T_{0} and {μ_{t}}_{t≥0} is a Brakke motion “sitting atop” the slices T_{t} of T. There can be a discrepancy between μ_{t} and μ_{T_{t}} due to weak lower semicontinuity of mass. The analogy in “straight” analysis is | lim Du| ≤ lim |Du| (as Radon measures).

8.1 Existence Theorem for Enhanced Motion. Let M^{n} be complete. Let T_{0} ∈ I_{k}^{loc}(M) satisfy 3.1, namely

ELLIPTIC REGULARIZATION 43

(i) ∂T_{0} = 0, M[T_{0}] < ∞. Then there exists T ∈ I_{k+1}^{loc}(M × [0, ∞)) and a family {μ_{t}}_{t≥0} of Radon measures, such that

(ii) (a) ∂T = T_{0}.

(b) M[T_{B}] is absolutely continuous with respect to L^{1}(B) for B ⊆ R. (iii) (a) μ_{0} = μ_{T_{0}}, M[μ_{t}] ≤ M[μ_{0}] for t > 0. (b) {μ_{t}}_{t≥0} is a Brakke motion.

(iv) μ_{t} ≥ μ_{T_{t}} for each t ≥ 0. The proof begins with 8.6. 8.2 Definition of Enhanced Motion. A pair (T, {μ_{t}}_{t≥0}) satisfying the conclu-

sions of 8.1 is called an enhanced motion with initial condition T_{0}.

We call T the undercurrent and {μ_{t}}_{t≥0} the overflow.

One advantage of this definition is that μ_{t} cannot arbitrarily disappear, as we see from 8.1(ii)(a), (b) and (iv). There still might be sudden mass loss.

The only reason for explicitly requiring M[μ_{t}] ≤ M[μ_{0}] in the definition is to rule out the creation of mass at infinity as in 3.6(b).

8.3 Additional Properties. The particular solution constructed in the proof is integral and satisfies, for B ⊆ R measurable,

M[T_{B}] ≤ (|B| + |B|^{1/2})M[T_{0}], M[π_{#}(T_{B})] ≤ |B|^{1/2}M[T_{0}].

It should be possible to prove this directly from definition 8.2, without recourse to the construction.

8.4 Restartability.

(i) By 8.1(iii)(a), (iv), we have

M[T_{t}] ≤ M[μ_{t}] ≤ M[μ_{0}] = M[T_{0}].

(ii) By the absolute continuity requirement 8.1(ii)(b), {T_{t}}_{t≥0} is continuous in the weak topology.

(iii) By 1.17, 1.15, (i), and (ii),

T_{t} ∈ I_{k}^{loc}(M × {t}) for each t ≥ 0.

(iv) Thus T_{t} satisfies the initial data hypothesis 8.1(i) for each t ≥ 0. This will come

in handy in §9. 8.5 Product Lemma. Let μ̄ be a Radon measure on U × (z_{0}, ∞), where U ⊆ M is

open. Suppose μ̄ is translation invariant in the sense that for all ψ ∈ C_{c}^{0}(U×(z_{0}, ∞), R^{+}) and all τ ≥ 0,

μ̄(ψ^{τ}) = μ̄(ψ)

where

ψ^{τ} (z) = \{ \begin{matrix}ψ(z − τ) & for z − τ > z_{0} \\ 0 & otherwise. \end{matrix}

Then

(i) If we choose θ ∈ C_{c}^{0}((z_{0}, ∞), R^{+}) such that ∫ θ(z) dz = 1 and define a Radon measure μ on U by

μ(ϕ) = μ̄(θϕ), ϕ ∈ C_{c}^{0}(U, R^{+}),

then μ is independent of θ, and

μ̄ = μ × (L^{1}⌊(z_{0}, ∞)).

(ii) If μ̄ ∈ M_{k+1}(U × (z_{0}, ∞)), then μ ∈ M_{k}(U). If μ̄ is integer rectifiable, so is μ. (iii) For any θ ∈ C_{c}^{2}((z_{0}, ∞), R^{+}) and any ϕ ∈ C_{c}^{2}(U, R^{+}),

B(μ̄, θϕ) = B(μ, ϕ).

Proof. We will assume that z_{0} = −∞, by extending μ̄ to a translation-invariant measure on all of U × R.

(i) Let K = {θ ∈ C_{c}^{0}(R, ∞) : ∫ θ dz = 1}. For each θ ∈ K, let K(θ) be the set of all finite convex combinations

θ′ = ∑ a_{τ} θ^{τ} , ∑ a_{τ} = 1, a_{τ} ≥ 0.

ELLIPTIC REGULARIZATION 45

By translation invariance, μ̄(θ′ϕ) is constant on K(θ). Two fixed functions θ_{1}, θ_{2} ∈ K can be approximated uniformly closely by elements of K(θ_{ε}) for a sufficiently tall, narrow θ_{ε}, proving μ̄(θ_{1}ϕ) = μ̄(θ_{2}ϕ) for all ϕ ∈ C_{c}^{0}(U, R^{+}), as required.

Then for ρ ∈ C_{c}^{0}(R, R^{+}), ϕ ∈ C_{c}^{0}(U, R^{+}),

μ̄(ρϕ) = μ(ϕ) L^{1}(ρ),

showing μ̄ = μ × L^{1}.

(ii) Assume μ̄ ∈ M_{k+1}(U × R). We will show μ ∈ M_{k}(U). First, note that T_{(x,z)}μ̄ ∋ ∂/∂z for μ̄-a.e. (x, z). If not, then by the rectifiability of μ̄, there exists (x, z) and a small C^{1} submanifold N^{k+1} ∋ (x, z) transverse to ∂/∂z such that μ̄(N) > 0. Translating N up and down the z-axis contradicts the local finiteness of μ̄.

By the rectifiability of μ̄, for μ̄-a.e. (x, z) ∈ U × R × [0, ∞) we have for all ϕ ∈ C_{c}^{0}(M, R^{+}),

lim_{λ↓0} μ_{x,λ}(ϕ) = lim_{λ↓0} μ̄_{(x,z),λ}(θϕ)
= Θ^{k+1}(μ̄, (x, z)) ∫_{T_{(x,z)}μ̄}θ(z)ϕ(x) dH^{k+1}(x, z)
= Θ^{k+1}(μ̄, (x, z)) ∫_{T_{(x,z)}μ̄∩R^{n}}ϕ(x) dH^{k}(x) since T_{(x,z)}μ̄ ∋ ∂/∂z,

so μ ∈ M_{k}(U) and T_{x}μ ⊕ Span (∂/∂z) = T_{(x,z)}μ̄ for μ-a.e. x and all z.

(iii) By the above, we see that μ⌊{ϕ > 0} is k-rectifiable if and only if μ̄⌊{ϕ > 0}×R is (k + 1)-rectifiable. In this case, let Y ∈ C_{c}^{1}(T ({ϕ > 0} × R)), then since μ̄ = μ × L^{1} and T_{(x,z)}μ̄ = T_{x}μ ⊕ Span (∂/∂z) μ-a.e., we obtain by the first variation formula,

δVμ̄(Y) = δV_{μ}(Ỹ),

where Ỹ = ∫ T_{(x,z)}μ · Y (x, z) dz. It follows that

δVμ̄⌊({ϕ > 0} × R) = δV_{μ}⌊{ϕ > 0} × L^{1}.

Then 6.2(ii)-(iv) hold for μ̄⌊({ϕ > 0} × R) if and only if they hold for μ⌊{ϕ > 0}. Also, for μ-a.e. x ∈ {ϕ > 0}

H⃗_{μ}(x) = H⃗μ̄(x, z) ⊥ ∂/∂z for all z ∈ R,

So in case 6.2(v) holds,

B(μ̄, θϕ) = ∫ −θϕH^{2} + H⃗ · S⊥ ; μ̄ · ¯∇(θϕ) dμ̄
= ∫ −ϕH^{2} + H⃗ · S_{μ}^{⊥} · ∇ϕ dμ + 0
= B(μ, ϕ).

This finishes Lemma 8.5. Q.E.D.

Proof of Theorem 8.1, consisting of 8.6-8.11.

8.6 Construction of Approximating Motion. For each ε > 0, let P^{ε} be the locally integral current of 3.2 satisfying

∂P^{ε} = T_{0},  I^{ε}[P^{ε}] ≤ M[T_{0}],
P^{ε} stationary for I^{ε} on (M × R) \ spt T_{0}.

Define

σ_{−t/ε}(x, z) = (x, z − \frac{t}{ε}),  P^{ε}(t) = (σ_{−t/ε})_{#} (P^{ε}),  μ_{t}^{ε} = μ_{P^{ε}(t)}.

8.7 Soliton Lemma. {μ_{t}^{ε}}_{t≥0} is an integral Brakke motion on the relatively open subset

W^{ε} ≡ \{(x, z, t) : z > −\frac{t}{ε}, t ≥ 0\}

of space-time (M × R) × [0, ∞). That is to say, μ^{ε} is a translative soliton, as asserted in §2.

Proof. Integrality is clear. Note that spt (∂P^{ε}(t)) × {t} lies outside of W^{ε}, so the following derivation is valid. Fix t ≥ 0 and ψ ∈ C_{c}^{2}(M, R^{+}) with spt ψ × {t} ⊂ W^{ε}. Recall from 2.6(i),(iii) that

S^{⊥}(x, z) · H⃗(x, z) = H⃗(x, z) = \frac{1}{ε}S^{⊥}(x, z) · ω

for μ^{ε}-a.e. (x, z). Inserting Y = ∂σ_{−t/ε}/∂t = −ω/ε in 2.4(1), we obtain

\frac{d}{dt}μ_{t}^{ε}(ψ) = \frac{d}{dt} ∫ ψ d(σ_{−t/ε})_{#}(μ^{ε})

= ∫ (−ψH⃗ + ∇ψ · S^{⊥}) · (−\frac{1}{ε}ω) dμ_{t}^{ε}

= ∫ −ψH^{2} + ∇ψ · S^{⊥} · H⃗ dμ_{t}^{ε} = B(μ_{t}^{ε}, ψ).

ELLIPTIC REGULARIZATION 47

Q.E.D.

Note that W^{ε} converges as ε ↓ 0 to W ≡ (M ×R×(0, ∞))∪(M ×(0, ∞)×{0}). We wish to pass to limits as ε ↓ 0 to obtain an integral Brakke motion {μ̄_{t}}_{t≥0} on W, where the domain of the Radon measure μ̄_{t} is M × R for t > 0, M × (0, ∞) for t = 0.

Fix t_{1} ≥ 0 and R_{1} < R_{2} such that R_{1} ≥ 0 if t_{1} = 0. Let

W_{2} = M × (R_{1}, R_{2}) × [t_{1}, ∞).

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 W_{2}(t_{1} = 0)  W_{2}(t_{1} > 0) W W^{ε}

Figure 7. Sets.

For ε small enough, W_{2} ⊆ W^{ε}. For t ≥ t_{1}, we have by 5.1 and 3.2(ii)

μ_{t}^{ε}(M × (R_{1}, R_{2})) ≤ (R_{2} − R_{1} + ε)M[T_{0}].

Hence the Brakke convergence theorem 7.1 applies to yield a subsequence ε_{i} ↓ 0 and an integral Brakke motion {μ̄_{t}}_{t≥0} on W such that, for all t ≥ 0,

(1)  μ_{t}^{ε_{i}} → μ̄_{t},  μ̄_{t}(M × (R_{1}, R_{2})) ≤ (R_{2} − R_{1})M[T_{0}].

For t = 0, the convergence is on M × (0, ∞).

8.8 Translation Invariance. Let t ≥ 0 and ψ ∈ C_{c}^{2}(M × R, R^{+}). If t = 0, assume spt ψ ⊆ C_{c}^{2}(M × (0, ∞), R^{+}). Then

(i) μ̄_{t}(ψ) ≥ μ̄_{t}(ψ^{τ}) ≥ lim_{s↓t} μ̄_{s}(ψ) for τ ≥ 0.

(ii) If t > 0, lim_{s↑t} μ̄_{s}(ψ) ≥ μ̄_{t}(ψ^{τ}) ≥ μ̄_{t}(ψ) for τ ≤ 0.

48 TOM ILMANEN (iii) In particular, by the continuity properties 7.2(i), (ii), we have

(σ_{τ})_{#}(μ̄_{t}) = μ̄_{t} for all τ ∈ R

for all but countably many t. Proof. Note that μ̄_{t}^{ε}(ψ^{τ}) = μ̄_{t+ετ}^{ε} (ψ). Recall from (1) and 7.2(i) that μ_{t}^{ε}(ψ)−C_{3}(ψ)t

is nonincreasing. Fix s > t and pass to limits as ε_{i} ↓ 0 to obtain

μ̄_{t}(ψ) − C_{3}(ψ)t ≥ μ̄_{t}(ψ^{τ}) − C_{3}(ψ)t ≥ μ̄_{s}(ψ) − C_{3}(ψ)s.

Therefore

μ̄_{t}(ψ) ≥ μ̄_{t}(ψ^{τ}) ≥ lim_{s↓t} μ̄_{s}(ψ)

as desired. (ii) is similar. (iii) is self-explanatory. Q.E.D.

8.9 Construction of {μ_{t}}_{t≥0}, proof of 8.1(iii)(b). Fix θ ∈ C_{c}^{2}((0, ∞), R^{+}) such that ∫_{−∞}^{∞} θ dz = 1. Define {μ_{t}}_{t≥0} by

μ_{t}(ϕ) = μ̄_{t}(θϕ), ϕ ∈ C_{c}^{2}(M, R^{+}).

Then (i) μ̄_{t} = μ_{t} × L^{1} for all but countably many t ≥ 0. (ii) {μ_{t}}_{t≥0} is an integral Brakke motion on M. Proof. (i) follows from lemma 8.8(iii) and the product lemma 8.5. (ii) Fix t_{0} ≥ 0 and ϕ ∈ C_{c}^{2}(M, R^{+}). It suffices to prove that both D_{t_{0}}^{−}μ_{t}(θϕ) and

D_{t_{0}}^{+}μ_{t}(θϕ) are less than B(μ_{t_{0}}, ϕ), where D^{−}, D^{+} are the left and right upper derivates. Let us treat the latter case, as the two cases are similar.

We can assume

D_{t_{0}}^{+}μ_{t}(θϕ) > −∞.

It follows that for any ψ̃ ∈ C_{c}^{2}({θϕ > 0}, R^{+}) with ψ̃ ≤ θϕ,

−∞ < D_{t_{0}}^{+} μ̄_{t}(θϕ) ≤ D_{t_{0}}^{+} μ̄_{t}(ψ̃) + C_{3}(θϕ − ψ̃),

so D_{t_{0}}^{+} μ̄_{t}(ψ̃) > −∞, and then by lemma 8.8(i),

μ̄_{t_{0}}(ψ̃^{τ}) = μ̄_{t_{0}}(ψ̃) for all τ ≥ 0.

ELLIPTIC REGULARIZATION 49

Now let z_{0} = min(spt θ) and let ψ ∈ C_{c}^{2}({ϕ > 0} × (z_{0}, ∞), R^{+}). Approximating ψ by a sum of translates ψ̃^{τ} , τ ≥ 0, of such functions ψ̃, we can prove

μ̄_{t_{0}}(ψ^{τ}) = μ̄_{t_{0}}(ψ) for all τ ≥ 0.

It follows by the product lemma 8.5(i) that

μ̄_{t_{0}}⌊({ϕ > 0} × (z_{0}, ∞)) = (μ_{t_{0}}⌊{ϕ > 0}) × (L^{1}⌊(z_{0}, ∞)).

Then using 8.5(iii),

D_{t_{0}}^{+}μ_{t}(ϕ) = D_{t_{0}}^{+} μ̄_{t}(θϕ) ≤ B(μ̄_{t_{0}}, θϕ) = B(μ_{t_{0}}, ϕ).

Together with the D^{−} case, this establishes 8.1(iii)(b). Note that 8.5(iii) also yields the integrality of {μ_{t}}_{t≥0}. Q.E.D.

8.10 Construction of T, proof of 8.1(ii) and 8.3. Let T^{ε} = (κ_{ε})_{#}(P^{ε}) as in 3.2. The mass estimate 5.3 and the I^{ε} bound 3.2(ii) give the mass bound needed for application of the current compactness theorem 1.17, yielding a subsequence of {ε_{i}}_{i≥1}, also to be called {ε_{i}}_{i≥1}, and a current T ∈ I_{k+1}^{loc}(M × R) such that

(i) T^{ε_{i}} ⇀ T, spt T ⊆ M × [0, ∞).

(ii) M[T_{B}] ≤ (|B| + |B|^{1/2})M[T_{0}] and M[π_{#}(T_{B})] ≤ |B| M[T_{0}] for all measurable

B ⊆ R, by 5.3.

8.11 Slice Convergence, proof of 8.1(iii)(a) and (iv), completion of 8.1.

(i) As ε_{i} ↓ 0, the family {T_{t}^{ε_{i}}}_{t≥0} converges uniformly in the flat metric topology

to a C^{1/2} path {T_{t}}_{t≥0} of currents T_{t} = ∂(T ⌊(M × [t, ∞))).

(ii) μ_{t} ≥ μ_{T_{t}} for all t ≥ 0. (iii) μ_{0} = μ_{T_{0}}, M[μ_{t}] ≤ M[μ_{0}]. Proof. (i) Let δ > 0 and choose η ∈ C^{∞}(R), 0 ≤ η ≤ 1, such that

η = \{ \begin{matrix}0 on (−∞, t − δ] \\ 1 on [t, ∞). \end{matrix}

Then for each t ≥ 0 and ϑ ∈ C_{c}^{∞}(Λ^{k}(M × R)),

lim_{i→∞} T^{ε_{i}}[t, ∞)(ϑ) = lim_{i→∞} T^{ε_{i}}[t, ∞)(ηϑ)

= lim_{i→∞} T^{ε_{i}}(ηϑ) − lim_{i→∞} T^{ε_{i}}[0, t)(ηϑ)

= T (ηϑ) ± (δ + δ^{1/2})M[T_{0}] max |ϑ| by 5.3.

Let δ ↓ 0 and obtain T^{ε_{i}}[t, ∞) ⇀ T [t, ∞) and thus, taking boundary,

T_{t}^{ε_{i}} ⇀ T_{t} for each t ≥ 0.

By 1.15, this implies convergence in the flat metric topology. It follows from the mass bound 5.3 and increasingly refined partitions of [0, ∞) that the convergence is uniform. We have (i).

(ii) Fix t ≥ 0 and let ϕ ∈ C_{c}^{2}(M, R^{+}). We seek to prove that μ_{t}(ϕ) ≥ μ_{T_{t}}(ϕ). By essentially the same argument as for (i), we have for each fixed τ,

T_{t+ε_{i}τ}^{ε_{i}} ⇀ T_{t},  lim inf_{ε_{i}↓0} μ_{T_{t+ε_{i}τ}^{ε_{i}}} (ϕ) ≥ μ_{T_{t}}(ϕ).

Then

μ_{t}(ϕ) = μ̄_{t}(θϕ)

= lim_{ε_{i}↓0} ∫ θ(z)ϕ(x) dμ_{t}^{ε_{i}}(x, z)

≥ lim inf_{ε_{i}↓0} ∫ θϕ|Dζ| dμ_{t}^{ε_{i}}

= lim inf_{ε_{i}↓0} ∫_{0}^{∞} θ ∫ ϕ dμ_{∂(P^{ε_{i}} (t)⌊M×[z,∞))}dz by co-area

= lim inf_{ε_{i}↓0} ∫_{0}^{∞} θ ∫ ϕ dμ_{P_{t/ε_{i}+z}^{ε_{i}}}dz by 8.6

= lim inf_{ε_{i}↓0} ∫_{0}^{∞} θ ∫ ϕ dμ_{T_{t+ε_{i}z}^{ε_{i}}}dz

≥ ∫_{0}^{∞} θ lim inf_{ε_{i}↓0} μ_{T_{t+ε_{i}z}^{ε_{i}}}(ϕ) dz

≥ μ_{T_{t}}(ϕ).

(iii) Recall from (1) that

μ̄_{t}(M × (R_{1}, R_{2})) ≤ (R_{2} − R_{1})M[T_{0}].

Approximating θ by step functions, this implies

M[μ_{t}] = ∫ dμ_{t} = ∫ θ(z) dμ̄_{t}(x, z) ≤ M[T_{0}].

Since μ_{0} ≥ μ_{T_{0}} by (ii), we also get μ_{0} = μ_{T_{0}}, as was desired for (iii). This completes Lemma 8.11 and Theorem 8.1. Q.E.D.

ELLIPTIC REGULARIZATION 51

§9 Matching Motion

The problem with the enhanced motion of the previous section is the discrepancy between μ_{t} and μ_{T_{t}}. One consequence is that an enhanced motion is not an enhanced motion with respect to later starting times.

It is a remarkable fact that under the hypothesis of uniqueness, the discrepancy is removable. On this observation rest the results of Chapter Two, culminating in the partial regularity result.

We define a matching motion to be an enhanced motion with zero discrepancy: 9.1 Definition of Matching Motion. Let M be complete and let T_{0} satisfy 3.1.

We call a current T ∈ I_{k+1}^{loc}(M × [0, ∞)) a matching motion for T_{0} provided (i) ∂T = T_{0}. (ii) M[T_{B}] ≤ (|B| + |B|^{1/2})M[T_{0}] for B ⊆ R measurable. (iii) {μ_{T_{t}}}_{t≥0} is a Brakke motion. If every enhanced motion pair (T, {μ_{t}}_{t≥0}) with initial condition T_{0} has the same

undercurrent T, then we say T is the unique undercurrent for T_{0}.

9.2 Existence of Matching Motion. Let M be complete and let T_{0} ∈ I_{k}^{loc}(M × {0}) be a cycle of finite mass. If T ∈ I_{k+1}^{loc}(M × [0, ∞)) is the unique undercurrent for T_{0}, then T is a matching motion.

Such uniqueness is not to be expected in codimension greater than one, as shown in the figure.

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Figure 8. Nonuniqueness.

52 TOM ILMANEN

Proof of 9.2. 1. Let (T, {μ_{t}}_{t≥0}) be an enhanced motion as constructed in §8. By the definition of enhanced motion, 9.1(i) already holds, and 9.1(ii) is true by 8.3.

2. Fix δ > 0. We begin by constructing an enhanced motion (T, {μ_{t}^{δ}}_{t≥0}) such that

M[μ_{t}^{δ}] < M[T_{t}] + δ for t ≥ 0.

Let Z_{δ} = {t : M[μ_{t}] ≥ M[T_{t}] + δ}. If Z_{δ} = ∅, we can take μ_{t}^{δ} = μ_{t} for all t ≥ 0. Otherwise, let t_{0} = inf Z_{δ}. By 8.4(ii), {T_{t}}_{t≥0} is weakly continuous, so M[T_{t}] is lower semicontinuous as a function of t. Also, M[μ_{t}] is nonincreasing, so we have

M[T_{t_{0}}] + δ ≤ lim inf_{t↓t_{0}} M[T_{t}] + δ ≤ lim_{t↓t_{0}} M[μ_{t}] ≤ M[μ_{t_{0}}]

so t_{0} ∈ Z_{δ}.

3. We now restart the evolution at time t_{0} with initial data T_{t_{0}}. (Recall from 8.4(iv) that T_{t_{0}} is legitimate initial data.) By 8.1 there is an enhanced motion (T̃, {μ̃_{t}}_{t≥t_{0}}) with

T̃ ∈ I_{k+1}^{loc}(M × [t_{0}, ∞)),  ∂T̃ = T_{t_{0}},  μ̃_{t_{0}} = μ_{T_{t_{0}}} ∈ M_{k}(M).

Now define

T̂ = T − T ⌊(M × [t_{0}, ∞)) + T̃,  μ̂_{t} = \{ \begin{matrix}μ_{t} & 0 ≤ t < t_{0} \\ μ̃_{t} & t_{0} ≤ t. \end{matrix}

Let us check that (T̂, {μ̂_{t}}_{t≥0}) is an enhanced motion for T_{0}. We have

∂T̂ = ∂T − ∂(T ⌊(M × [t_{0}, ∞))) + ∂T̃
= T_{0} − T_{t_{0}} + T_{t_{0}}
= T_{0}.

so T̂ satisfies 8.1(ii)(a). In addition, the absolute continuity requirement 8.1(ii)(b) holds for T̂ as it did for T, T̃ separately. Evidently we also have

8.1(iii)(a) : μ̂_{0} = μ_{T_{0}},
8.1(iv) : μ̂_{t} ≥ μT̂_{t} for all t ≥ 0.

Let us check 8.1(iii)(b), namely that {μ̂_{t}}_{t≥0} is a Brakke motion. This is clear for t ≠ t_{0} and for t ↓ t_{0}. Recall

D_{t_{0}}^{−}f(t) = lim sup_{t↑t_{0}} \frac{f(t) − f(t_{0})}{t − t_{0}}.

We must check that D_{t_{0}}^{−}μ̂_{t}(ϕ) ≤ B(μ̂_{t_{0}}, ϕ) for ϕ ∈ C_{c}^{2}(M, R^{+}). Recall that

μ̂_{t_{0}} = μ̃_{t_{0}} = μ_{T_{0}} ≤ μ_{t_{0}}.

If μ̂_{t_{0}}(ϕ) = μ_{t_{0}}(ϕ), then

μ̂_{t_{0}}⌊{ϕ > 0} = μ_{t_{0}}⌊{ϕ > 0}

and

D_{t_{0}}^{−}μ̂_{t}(ϕ) = D_{t_{0}}^{−}μ_{t}(ϕ) ≤ B(μ_{t_{0}}, ϕ) = B(μ̂_{t_{0}}, ϕ)

by 6.4(b). On the other hand, if μ̂_{t_{0}}(ϕ) < μ_{t_{0}}(ϕ), then by the semidecreasing property 7.2(i), D_{t_{0}}^{−}μ̂_{t}(ϕ) = −∞ and we have D_{t_{0}}^{−}μ̂_{t}(ϕ) ≤ B(μ̂_{t_{0}}, ϕ) in this case as well.

In summary, we have proven that (T̂, {μ̂_{t}}_{t≥0}) is an enhanced motion for T_{0}.

4. Therefore, by the uniqueness hypothesis of the theorem, we have T̂ = T. As in step 2, we argue

lim_{t↓t_{0}} M[μ̂_{t}] ≤ M[μ̂_{t_{0}}] = M[T_{t_{0}}] ≤ lim inf_{t↓t_{0}} M[T_{t}]

from which it follows that we have increased the value of inf Z_{δ}. Notice also that

M[μ̂_{t_{0}}] ≤ M[μ_{t_{0}}] − δ ≤ M[μ̂_{0}] − δ.

Therefore, if we repeat the process, it must terminate after no more than [M[T_{0}]/δ] iterations. We have constructed an enhanced motion (T, {μ_{t}^{δ}}_{t≥0}) with M[μ_{t}^{δ}] < M[T_{t}] + δ for all t ≥ 0.

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Figure 9. Restarting Procedure.

5. By the Brakke convergence theorem 7.1 there is a Brakke motion {μ′_{t}}_{t≥0} with

μ_{t}^{δ_{i}} → μ′_{t}, μ′_{t} ≥ μ_{T_{t}} for all t ≥ 0,

and by step 4, M[μ′_{t}] = M[T_{t}] for each t ≥ 0, so actually,

μ′_{t} = μ_{T_{t}} for all t ≥ 0.

Thus we have shown that {μ_{T_{t}}}_{t≥0} is a Brakke motion, establishing 9.2. Q.E.D.

Remark. The construction of μ^{δ} can be done for each δ without the uniqueness assumption, at the cost of changing T for each δ. But in passing to limits, there could be cancellation of mass.

Chapter Two. Partial Regularity in Codimension One.

Codimension one surfaces have the special property that they avoid each other when moving by mean curvature. That is to say, the distance between disjoint surfaces never decreases. Therefore we can fill space with a family of disjoint surfaces, indexed as the level-sets of a Lipschitz function u, as in the work of Evans-Spruck [ES1] and Chen-Giga-Goto [CGG]. Because the level-sets do not approach one another, u(·, t) remains Lipschitz.

Moreover, we can prove that the support of the enhanced motion (T, {μ_{t}}_{t≥0}) defined in §8 remains within the corresponding level set of u. Necessarily, all but countably many level-sets of u have zero L^{n+1}-measure in space-time. For such a “thin” level-set, the current T of 8.2 is uniquely determined as the boundary (in the sense of currents) of the set {u > a}. It follows by Theorem 9.2 that T is a matching motion, that is to say, {μ_{T_{t}}}_{t≥0} is a Brakke motion.

Furthermore, we will show that T_{t} = ∂[{x : u(x, t) > a}]. Then by DeGiorgi’s theorem on sets of finite perimeter, μ_{T_{t}} will have unit density μ_{T_{t}}-a.e. This, in turn, provides the crucial hypothesis for the application of Brakke’s powerful Local Regularity Theorem to the motion. As a result, we will prove the following:

Let T_{0} be the reduced boundary of a open set of finite perimeter in R^{n}. If the level-set flow of the support of T_{0} does not develop positive Lebesgue measure, then T_{0} has a unique undercurrent T, whose time-slices form a unit density Brakke motion. Furthermore, spt T is smooth H^{n}-almost everywhere.

As a corollary, if u_{0} is a Lipschitz function with compact level-sets and u is the corresponding level-set flow, then almost every level-set of u is smooth H^{n}-almost everywhere.

Recently Evans and Spruck [ES4] have discovered an unsuspected global estimate on the integral of |H|, and have used it to improve weak to strong convergence for almost every level of the level-set flow, yielding unit density. This constitutes an alternate pathway to the application of Brakke’s regularity theory to the level-set flow.

The organization of the chapter is as follows. In §10 we prove that a Brakke motion never collides with a manifold moving smoothly by mean curvature. It follows by the geometric avoidance principle of [I2] that two Brakke motions, initially disjoint, never collide, and that a Brakke motion remains included within the corresponding level-set 56 TOM ILMANEN

motion. In §11, we show that a level-set that doesn’t fatten up has a unique undercurrent which is a unit density Brakke motion. In §12, we apply Brakke’s Local Regularity Theorem to yield theorems about almost-everywhere regularity for motion by mean curvature.

For simplicity, take M = R^{n}.

§10 Avoidance and Inclusion for Brakke Motions

Our first task in this section is to sketch a geometric characterization of the levelset flow. Soner did this originally in [So], using the evolving distance function. Our approach, using smooth barrier surfaces, appears in [I2]. It differs slightly from [So] because it applies to (unoriented) closed sets rather than to topological boundaries.

The second step is to show that a Brakke motion avoids these smooth barriers. This is inspired by [B, 3.6]; see also the varifold maximum principle of Solomon-White [SW]. In consequence, a Brakke motion that is initially included in a level-set moving by mean curvature, remains included in it.

10.1 Set-Theoretic Subsolution. ([I2, def. 4A]) A family {Δ_{t}}_{t≥0} of closed sets in R^{n} is said to be a set-theoretic subsolution of the mean curvature flow provided that for any family {Q_{t}}_{t∈[t_{0},t_{1}]} of smooth, compact hypersurfaces moving by mean curvature,

Δ_{t_{0}} ∩ Q_{t_{0}} = ∅  implies  Δ_{t} ∩ Q_{t} = ∅ for all t ∈ [t_{0}, t_{1}]

or equivalently,

dist(Δ_{t}, Q_{t}) ≥ dist(Δ_{t_{0}}, Q_{t_{0}}) for t ∈ [t_{0}, t_{1}].

Proof of Equivalence. Clearly the second condition implies the first. For the converse, we use the translation invariance of motion by mean curvature. If dist(Δ_{t}, Q_{t}) < dist(Δ_{t_{0}}, Q_{t_{0}}) ever occurs, then some translate of {Q_{t}}_{t∈[t_{0},t_{1}]} violates 10.1. Q.E.D.

The justifying virtue of set-theoretic subsolutions is that they satisfy the maximum principle with respect to each other.

ELLIPTIC REGULARIZATION 57

10.2 Avoidance Lemma for Set-Theoretic Subsolutions. ([I2, Lemma 4E]) Let {Δ_{t}}_{t≥0}, {Γ_{t}}_{t≥0} be set-theoretic subsolutions of the mean curvature flow in R^{n}. Assume

Δ_{0} ∩ Γ_{0} = ∅, Γ_{0} compact.

Then dist(Δ_{t}, Γ_{t}) is nondecreasing.

The proof (see [I2]) relies on the translation invariance of the equation, and should apply to “viscosity solutions” in other contexts as well. The basic idea is to interpose a C^{1,1} surface between Γ_{t} and Δ_{t}, then use the fact that set-theoretic subsolutions avoid smooth flows.

10.3 Relation to Level-Set Flow. (see [I2]) Let us relate the concept of settheoretic subsolutions to the level-set flow introduced by Evans-Spruck and Chen-Giga-Goto. In [ES1] and [CGG], the following equation appears together with a (viscosity) definition of its weak solutions.

(⋆)  \{ \begin{matrix}\frac{∂u}{∂t} & = (δ − ν ⊗ ν) : ∇^{2}u  on R^{n} × [0, ∞) \\ u(·, 0) & = f(·),  along R^{n} × \{0\}, \end{matrix}

where δ is the identity matrix and ν = Du/|Du|. When u is smooth and Du ≠ 0, equation (⋆) says that the level-sets of u are moving by mean curvature. If f is continuous and all but at most one of the level-sets of f are compact, then there exists a unique u weakly solving (⋆) (see [I1, §7]). The family of level-sets Γ_{t}^{a} ≡ {x : u(x, t) = a}, t ≥ 0, is unique and is called the level-set flow (by mean curvature) of f^{−1}(a). It follows trivially from the definition of weak solution of (⋆) (which involves tangency of u with smooth test functions, see [ES1]) that a level-set flow in R^{n} is a set-theoretic subsolution of mean curvature motion.

In fact, any set-theoretic subsolution that is contained within a level-set flow at t = 0 must remain contained within it, because otherwise it would run into some of the other level-sets of u, violating 10.2. (One of the two sets involved in the collision is compact by the hypothesis on f). This proves

10.4 Equivalence of Definitions. A level-set flow in R^{n} is a maximal settheoretic subsolution of mean curvature motion in the sense that it contains all other set-theoretic subsolutions with the same initial condition.

58 TOM ILMANEN

Let us now briefly sketch how to derive the existence, uniqueness and properties of u and of the level-set flow directly from 10.2, alternate to the methods of Evans-Spruck, Chen-Giga-Goto, and Soner; most of this was done in [I2]. Let f be a continuous function on R^{n} with all level-sets compact (except possibly one). For each a ∈ R, we can instantly construct the (unique) maximal set-theoretic subsolution {Γ_{t}^{a}}_{t≥0} with initial condition f^{−1}(a) by

Γ_{t}^{a} = \overline{⋃\{Δ_{t} : \{Δ_{s}\}_{s≥0} satisfies 10.1, Δ_{0} = f^{−1}(a)\}}.

(Evidently {Γ_{t}}_{t≥0} satisfies the second equivalent property of 10.1.) By 10.2, the {Γ_{t}^{a}}_{t≥0} are all disjoint. In fact, the maximality property implies, with some work, that for each fixed t, the sets Γ_{t}^{a}, a ∈ R, form a partition of space. Using 10.1, it is then easy to show that they constitute the level-sets of a continuous function u weakly solving (⋆), establishing the existence and uniqueness of u. Therefore, once again, the maximal settheoretic subsolution is a level-set flow and vice versa.

Having outlined prior art, let us turn to Brakke motions. The main result of this section is

10.5 Tangency Property of Brakke Motions. The support of any codimension one Brakke motion on R^{n} is a set-theoretic subsolution of motion by mean curvature.

Then by 10.2 we obtain immediately

10.6 Avoidance Lemma for Codimension One Brakke Motions. Let {μ_{t}}_{t≥0}, {ν_{t}}_{t≥0} be codimension one Brakke motions on R^{n} such that spt ν_{0} is compact and

spt μ_{0} ∩ spt ν_{0} = ∅.

Then dist(spt μ_{t}, spt ν_{t}) is nondecreasing.

By 10.4 we have

10.7 Inclusion Theorem. Let {μ_{t}}_{t≥0} be a codimension one Brakke motion on R^{n} and let {Γ_{t}}_{t≥0} be a level-set flow. Then

spt μ_{0} ⊆ Γ_{0} implies spt μ_{t} ⊆ Γ_{t} for all t ≥ 0.

Proof of 10.5. Let {μ_{t}}_{t≥0} be a Brakke motion and let {Q_{t}}_{t∈[t_{0},t_{1}]} be a family of compact hypersurfaces flowing smoothly by mean curvature, such that spt μ_{t_{0}} ∩ Q_{t_{0}} = ∅. The idea is to construct a test function for μ_{t} out of the distance function to Q_{t}.

1. Let E_{t} be the compact region bounded by Q_{t} and define

r(x, t) = \{ \begin{matrix}− & dist(x, Q_{t})  x ∈ E_{t} \\ dist(x, Q_{t})  x ∉ E_{t}. \end{matrix}

Fix γ > 0 so small that

dist(spt μ_{t_{0}}, Q_{t_{0}}) > γ

and such that r(x, t) is smooth on the set

U ≡ {(x, t) : −γ < r(x, t) < γ, t_{0} ≤ t ≤ t_{1}}.

Let β > 2 be a number to be selected later. Define the test function

ϕ = h(r) ≡ \{ \begin{matrix}(γ − |r|)^{β} & |r| ≤ γ \\ 0 & |r| ≥ γ. \end{matrix}

= 0.880cm  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  γ −γ  0

Figure 10. Graph of ϕ.

Then ϕ ∈ C_{c}^{0}(R^{n} × [t_{0}, t_{1}]), ϕ vanishes except on U, and ϕ is C^{2} except along Q_{t}.

Observe that μ_{t_{0}}(ϕ) = 0. We aim to show this remains so for t ∈ [t_{0}, t_{1}]. This will complete the proof.

2. Let t_{∗} = sup{t ∈ [t_{0}, t_{1}] : μ_{t}(ϕ) = 0}. Assume, for a contradiction, that t_{∗} < t_{1}. By the growth bound 6.7, we have μ_{t_{∗}}(ϕ) = 0. Thus dist(spt μ_{t_{∗}}, Q_{t_{∗}}) ≥ γ, so spt μ_{t_{∗}} is disjoint from any ball B_{γ/2}(x) centered at a point x ∈ Q_{t_{∗}}. Now let B_{γ/2}(x) begin to flow by mean curvature. By Brakke’s Internal Sphere Barrier Lemma [B, 3.6], spt μ_{t} remains disjoint from the evolute of B_{γ/2}(x) for as long as the latter exists. So there is τ > 0 such that

spt μ_{t} ∩ Q_{t} = ∅ for t ∈ [t_{∗}, t_{∗} + τ].

60 TOM ILMANEN

Therefore ϕ is C^{2} on spt μ_{t}.

3. Thus, for t ∈ [t_{∗}, t_{∗} +τ], we can legitimately insert ϕ into the time-varying Brakke inequality 6.9 to obtain, by completing the square and using the first variation formula 1.8,

D̄_{t}μ_{t}(ϕ) ≤ ∫ −ϕH^{2} + ∇ϕ · S^{⊥} · H⃗ + \frac{∂}{∂t}ϕ dμ_{t}

= ∫ −ϕH^{2} − ∇ϕ · S · H⃗ + ∇ϕ · H⃗ + \frac{∂}{∂t}ϕ dμ_{t}

= ∫ − |ϕ^{1/2}H⃗ + \frac{1}{2} \frac{S · ∇ϕ}{ϕ^{1/2}} |^{2} + \frac{1}{4} \frac{|∇ϕ · S|^{2}}{ϕ} − S : ∇^{2}ϕ + \frac{∂}{∂t}ϕ dμ_{t}

≤ ∫ \frac{1}{4} \frac{|∇ϕ · S|^{2}}{ϕ} − S : ∇^{2}ϕ + \frac{∂}{∂t}ϕ dμ_{t}.

(If {μ_{t}}_{t≥0} is integral, we can use Brakke’s result [B, 5.8] that S · H⃗ = 0 μ_{t}-a.e. to drop a term, but the proof doesn’t become any easier.) Plug in ϕ = h(r),

∇ϕ = h′(r)∇r,  ∇^{2}ϕ = h′′(r)∇r ⊗ ∇r + h′(r)∇^{2}r,  \frac{∂}{∂t}ϕ = h′(r)\frac{∂}{∂t}r

to get

D̄_{t}μ_{t}(ϕ) ≤ ∫ (\frac{1}{4} \frac{(h′)^{2}}{h} − h′′) |∇r · S|^{2} + h′ (−S : ∇^{2}r + \frac{∂}{∂t}r) dμ_{t}.

Since {Q_{t}}_{t∈[t_{0},t_{1}]} solves the smooth mean curvature flow, a standard calculation (see [ES2]) tells us that

r (\frac{∂}{∂t}r − Δr) ≥ 0,

on U, where Δ = Δ^{R^{n}}. Since h′(r)r ≤ 0, we have

h′(r) (\frac{∂}{∂t}r − Δr) ≤ 0.

Now Δr = S : ∇^{2}r + n⃗ · ∇^{2}r · n⃗, where n⃗ is the unit normal of S, and therefore

D̄_{t}μ_{t}(ϕ) ≤ ∫ (\frac{1}{4} \frac{(h′)^{2}}{h} − h′′) (1 − |∇r · n⃗|^{2}) + h′n⃗ · ∇^{2}r · n⃗ dμ.

Define the (n − 1)-plane field T (x) on U by T = (∇r)^{⊥}, note that n⃗ = n⃗ · T + (n⃗ · ∇r)∇r, and obtain

\begin{aligned}D̄_{t}μ_{t}(ϕ) & ≤ ∫ (\frac{1}{4} \frac{(h′)^{2}}{h} − h′′) |n⃗ · T |^{2} + |h′||n⃗ · ∇^{2}r · n⃗| dμ_{t} \\ & = ∫ (\frac{1}{4} \frac{(h′)^{2}}{h} − h′′ + |h′||T · ∇^{2}r · T |) |n⃗ · T |^{2} dμ_{t} \end{aligned}

where we have used the fact that ∇r · ∇^{2}r = 0, as can be seen by differentiating |∇r|^{2} = 1. Now substitute

\begin{aligned}h & = (γ − |r|)^{β}, \\ h′ & = −sgn (r)β(γ − |r|)^{β−1}sgn r, \\ h′′ & = β(β − 1)(γ − |r|)^{β−2}, \\ \frac{(h′)^{2}}{h} & = β^{2}(γ − |r|)^{β−2}, \end{aligned}

valid on spt μ_{t}, t ∈ [t_{0}, t_{1}], to obtain

D̄_{t}μ_{t}(ϕ) ≤ ∫_{U} β(γ − |r|)^{β−2}|n⃗ · T |^{2} [1 − \frac{3}{4}β + (γ − |r|)|T · ∇^{2}r · T |] dμ_{t}.

Now choose β ≥ (4/3)(1 + γ(n − 1) max\overline{U} |∇^{2}r|). Notice that β is independent of t and t_{∗}. (Alternately fix β = 3 and choose γ so that γ(n − 1) max\overline{U} |∇^{2}r| < 5/4.) Then

D̄_{t}μ_{t}(ϕ) ≤ 0 for all t ∈ [t_{∗}, t_{∗} + τ].

and therefore μ_{t}(ϕ) = 0 for t ∈ [t_{∗}, t_{∗} + τ], contradicting the definition of t_{∗}. Hence t_{∗} = t_{1}, and μ_{t}(ϕ) = 0 for all t ∈ [t_{0}, t_{1}], proving 10.5. Q.E.D.

§11 Unit Density

We will see that the inclusion theorem of the last chapter, together with the matching theorem 9.2, shows that a solution of the level-set flow that does not fatten up corresponds to a unique Brakke motion of unit density that is the boundary of a set of finite perimeter in space-time.

Define

τ(θ) = τ({θ ≠ 0}, |θ|, sgn(θ) e_{1} ∧ … ∧ e_{p})

whenever θ ∈ L_{loc}^{1}(R^{p}, Z) is a function of locally bounded variation. Then τ(θ) ∈ I_{p}^{loc}(R^{p}). The perimeter of a set E ⊆ R^{p} is M[∂[E]], where [E] ≡ τ(χ_{E}).

11.1 Relation of Enhanced Motion to the Level-Set Flow.

(i) Let E_{0} ⊆ R^{n} be a bounded open set of finite perimeter, and let T_{0} = ∂[E_{0}]. Then there is a set E ⊆ R^{n} × [0, ∞) of finite perimeter and an enhanced motion (T, {μ_{t}}_{t≥0}) such that

T = ∂[E] + [E_{0}], T_{t} = ∂[E_{t}],

62 TOM ILMANEN

where E_{t} = E ∩ (M × {t}), and {χ_{π_{#}(E_{t})}}_{t≥0} is absolutely continuous in L^{1}(R^{n}).

(ii) Let E, (T, {μ_{t}}_{t≥0}) be objects satisfying (i). Let f : R^{n} → R be a continuous function with E_{0} = {f > 0}, and let u be the unique continuous function weakly solving the level-set flow equation (⋆) of §10. Then E may be modified on a set of L^{n+1}-measure zero, without affecting the properties of (i), so that

{u > 0} ⊆ E ⊆ {u ≥ 0}.

We will prove this after 11.7.

11.2 Definition of Boundary Motion. Let E_{0} be a bounded open set of finite perimeter in R^{n} with T_{0} = ∂[E_{0}]. A current T ∈ I_{n}(M × [0, ∞)) is called a boundary motion for T_{0} provided there is a relatively open set E ⊆ R^{n} × [0, ∞) of finite perimeter such that

(i) T = ∂[E] + [E_{0}], T_{t} = ∂[E_{t}] where E_{t} = E ∩ (M × {t}), and {χ_{π_{#}(E_{t})}}_{t≥0} is absolutely continuous in L^{1}(R^{n}).

(ii) {μ_{T_{t}}}_{t≥0} is a Brakke motion.

In particular, each E_{t} is a set of finite perimeter. DeGiorgi’s theorem (discussed in §12) implies that μ_{∂[E_{t}]} has unit density almost everywhere. Since this is the crucial requirement for Brakke’s Regularity Theory, a boundary motion is a very desirable kind of mean curvature motion.

11.3 Definition of Nonfattening. Let {Γ_{t}}_{t≥0} be a level-set flow. We say that {Γ_{t})_{t≥0} satisfies the nonfattening condition provided that

L^{n+1}(⋃_{t≥0} Γ_{t} × {t}) = 0

Remark. If H^{n−1}(Γ_{0}) < ∞, then the following are equivalent forms of the nonfattening hypothesis:

(i) L^{n+1}( ⋃_{t≥0} Γ_{t} × {t}) = 0.

(ii) ⋃_{t≥0} Γ_{t} × {t} has no (space-time) interior.

(iii) L^{n}(Γ_{t}) = 0 for a.e. t ≥ 0.

(iv) For a.e. t ≥ 0, Γ_{t} has no spatial interior.

ELLIPTIC REGULARIZATION 63

(v) L^{n}(Γ_{t}) = 0 for all t ≥ 0.

(vi) For all t ≥ 0, Γ_{t} has no spatial interior.

Proof of Equivalence. Clearly (i) implies (ii), (iii) implies (iv), and (v) implies (vi). By Evans-Spruck [ES3, 4.2], H^{n−1}(∂Γ_{t}) ≤ CH^{n−1}(Γ_{0}) < ∞ for all t ≥ 0. Thus in fact

(iii) is equivalent to (iv), (v) is equivalent to (vi).

Furthermore, if Γ_{t} develops an interior, it retains it for a short time (by considering a shrinking ball embedded in Γ_{t}). Thus

(ii), (iv) and (vi) are equivalent.

Finally, (i) is equivalent to (iii).

by Fubini’s theorem, since ⋃_{t≥0} Γ_{t} × {t} = {u = 0} is closed. Q.E.D.

11.4 Unit Density Theorem. Let E_{0} ⊆ R^{n} be a bounded open set of finite perimeter, T_{0} = ∂[E_{0}]. Let f be any continuous, proper function with E_{0} = {f > 0}, ∂E_{0} = {f = 0}. Let u solve (⋆) for f, and let {Γ_{t}}_{t≥0} be the corresponding level-set flow. Define E = {u > 0}, T = ∂[E] + [E_{0}].

If {Γ_{t}}_{t≥0} satisfies the nonfattening condition, then T is the unique undercurrent for T_{0}, T is a boundary motion, and {μ_{T_{t}}}_{t≥0} is a unit density Brakke motion.

Only a countable number of level-sets of a given function can fatten up, so any level-set can be perturbed to one satisfying the above theorem.

Remark. The nonfattening hypothesis is always fulfilled in the following special cases:

(a) T_{0} is smooth of positive mean curvature.

(b) T_{0} bounds a smooth star-shaped region (see Soner [So]). See Soner-Souganidis [SS] for other examples. To proceed, we will need two lemmas.

11.6 Support Lemma. Let T be the undercurrent for an enhanced motion of dimension k in R^{n}. Then, with D = D^{T}, (i) The set Z ≡ {(x, t) : |Dτ(x, t)| is undefined or zero} has μ_{T} (Z) = σ(Z) = 0,

where σ is given by

σ(ϕ) = ∫∫ ϕ dμ_{T_{t}}dt, ϕ ∈ C_{c}^{0}(M × [0, ∞), R^{+}).

(ii) μ_{T} and σ are mutually absolutely continuous.

(iii) spt T = \overline{⋃_{t≥0} spt T_{t}}.

Proof. (i) Let Z_{0} = {(x, t) : |Dτ(x, t)| = 0}. We will prove μ_{T} (Z_{0}) = 0. (Note that σ(Z_{0}) = 0 is trivial by the co-area formula.) Let T = τ(X, θ, ξ). By the rectifiability of X, we have

Z_{0} ⊆ X ⊆ ⋃_{i≥1} C_{i} H^{k+1}-a.e.

where C_{i} is a C^{1} (k+1)-submanifold and |D_{C_{i}}τ| = |Dτ| = 0 for H^{k+1}-a.e. (x, t) ∈ C_{i}∩Z_{0}. Thus by Sard’s theorem for C^{1} functions [S, 10.4] (an immediate consequence of the coarea formula)

Z_{0} = Z_{1} ∪ Z_{2}

where H^{k+1}(Z_{1}) = 0 and L^{1}(τ(Z_{2})) = 0. Then μ_{T} (Z_{1}) = 0, and applying the absolute continuity 8.1(ii)(b), we have μ_{T} (Z_{2}) = 0. Thus μ_{T} (Z_{0}) = 0.

If Z_{∗} is the set where |Dτ| is undefined, then μ_{T} (Z_{∗}) = 0. Therefore μ_{T} (Z) = μ_{T} (Z_{0} ∪ Z_{∗}) = 0.

(ii) By the co-area formula, for any W

σ(W) = ∫_{W} |Dτ| dμ_{T} .

Since |Dτ| ≠ 0 μ_{T} -a.e., σ and μ_{T} are mutually absolutely continuous. In particular σ(Z) = 0.

(iii) Clearly spt T_{t} ⊆ spt T for each t. For the other direction, let (x, t) ∈ spt T and let U ∋ (x, t) be open. Then μ_{T} (U) > 0, so by (ii), σ(U) > 0, i.e.

∫_{0}^{∞} ∫ χ_{U} dμ_{T_{s}}ds > 0.

Thus M[T_{s}⌊U] > 0 for some s near t. Since U was arbitrary, (x, t) ∈ \overline{⋃_{s≥0} spt T_{s}}.

11.7 Impossibility of Spontaneous Generation. Let u : R^{n} × [0, ∞) → R be a continuous function solving §10(⋆). Then each component of {u ≠ 0} meets R^{n} × {0}.

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 R^{n} × {0}

Figure 11. Zero set of u.

Proof. Let U be a component of {u ≠ 0}. By applying the relabelling lemma [I1, 3.2] locally, it is evident that the function u^{ε} defined by

u^{ε} = \{ \begin{matrix}u & on \{−ε ≤ u ≤ ε\} ∪ U \\ ε & on \{ε ≤ u\} \ U \\ −ε & on \{u ≤ −ε\} \ U \end{matrix}

will solve (⋆). Passing to limits by the uniform convergence lemma [I1, 3.1], we obtain ũ solving (⋆) with U = {ũ ≠ 0}. If U ∩ (R^{n} × {0}) = ∅, then we are contradicting uniqueness of the zero solution of (⋆). Q.E.D.

Proof of 11.1. 1. Let E_{0} ⊆ R^{n} be a bounded open set of finite perimeter and let T_{0} = ∂[E_{0}]. Let (T, {μ_{t}}_{t≥0}) be the enhanced motion with initial condition T_{0} constructed in the proof of 8.1.

It is not hard to check that the existence procedure for the current T, which takes place in 3.2 and §8, can also be made to yield a set E ⊆ R^{n} × [0, ∞) of finite perimeter such that

T = ∂[E] + [E_{0}].

One obtains this result by initially restricting the admissible set A of 3.2 to consist only of currents of this form, and carrying out convergence of the characteristic function of E in L^{1}(R^{n} × [0, ∞)) by means of the compactness theorem for BV_{loc}.

66 TOM ILMANEN

2. Let us rename E_{0} as E_{0}^{given}. Define E_{t} = E ∩ (R^{n} × {t}), t ≥ 0. (We will worry about consistency of E_{0} with E_{0}^{given} shortly.) By the slice theorem [S, §28] we have for a.e. 0 < t < s,

(1)  \begin{aligned}∂([E]⌊(R^{n} × [t, s))) & = ∂[E]⌊(R^{n} × [t, s)) + ⟨[E], τ, s⟩ − ⟨[E], τ, t⟩ \\ & = (T − [E_{0}^{given}])⌊(R^{n} × [t, s)) + [E_{s}] − [E_{t}] \\ & = T [t, s) + [E_{s}] − [E_{t}]. \end{aligned}

Project onto R^{n} to obtain, since π_{#}([E]⌊(R^{n} × [t, s))) = 0,

0 = π_{#}∂([E]⌊(R^{n} × [t, s))) = π_{#}T [t, s) + π_{#}[E_{s}] − π_{#}[E_{t}].

By the absolute continuity of M[T (t, s)] with respect to |t − s| (see 8.1(ii)(b)), we find that

M[π_{#}[E_{s}] − π_{#}[E_{t}]] ↓ 0 as |s − t| ↓ 0

for the full measure set of s, t such that (1) holds. Thus we may change E by a set of L^{n+1}-measure zero so that

{χ_{π(E_{t})}}_{t≥0}

is an absolutely continuous path in L^{1}(R^{n}).

3. Setting s = ∞ and taking the boundary of (1) we obtain

T_{t} = ∂(T [t, ∞)) = ∂[E_{t}] a.e. t > 0.

Since both {T_{t}}_{t≥0} and {[E_{t}]}_{t≥0} are weakly continuous, we obtain

T_{t} = ∂[E_{t}] for all t ≥ 0.

Now T_{0} = ∂[E_{0}^{given}], so ∂([E_{0}] − [E_{0}^{given}]) = 0, which shows by the Constancy Theorem [S, 26.27] that the two definitions of E_{0} are equal up to a set of L^{n}-measure zero. This completes the proof of assertion 11.1(i).

4. Let f, u be as in hypothesis 11.1(ii). In particular spt μ_{0} is compact and contained in {f = 0}. By 11.6(iii), the definition of enhanced motion, and the inclusion lemma 10.7,

spt T = \overline{⋃_{t≥0} spt T_{t}} ⊆ \overline{⋃_{t≥0} spt μ_{t} × \{t\}} ⊆ \{u = 0\}.

ELLIPTIC REGULARIZATION 67

Thus spt ∂E = spt (T − [E_{0}]) ⊆ {u = 0} ∪ {f ≥ 0}.

Now let U be any component of {u > 0} relative to R^{n} × [0, ∞). By the Constancy Theorem and the above, χ_{E} is constant L^{n+1}-a.e. on U. Since {χ_{E_{t}}}_{t≥0} is continuous in L^{1} and U is open relative to R^{n} × [0, ∞), we can redefine E on a set of L^{n+1}-measure zero, and each E_{t} on a set of L^{n}-measure zero without affecting the continuity in L^{1}, so that

χ_{E} is constant on U.

By the Spontaneous Generation Lemma 11.7, U meets R^{n} × {0} in one or more components of {f > 0} = E_{0}, so in fact U ⊆ E. Since U was arbitrary, it follows that {u > 0} ⊆ E.

A similar argument shows that {u < 0} ⊆ R^{n} × [0, ∞) \ E. Thus E is trapped between {u > 0} and {u ≥ 0}. This is assertion 11.1(ii). Q.E.D.

Proof of 11.4. 1. Let T be the current constructed in 11.1 and let T^{1} be any other undercurrent for T_{0}, with corresponding Brakke motion {μ_{t}^{1}}_{t≥0}. As in the previous proof, we find that

spt T^{1} ⊆ {u = 0}.

Then spt (T^{1} − T) ⊆ {u = 0}. Also, ∂(T^{1} − T) = T_{0} − T_{0} = 0, so by homology T^{1} − T = ∂F for some current F ∈ I_{n+1}^{loc}(R^{n} × R) with spt ∂F ⊆ {u = 0}. By 11.7, the set {u = 0} (which lies in R^{n} × [0, ∞)) does not separate R^{n} × R. Hence F = τ(θ) for a function θ that is almost everywhere constant, in fact zero, on {u ≠ 0}. By hypothesis L^{n+1}({u = 0}) = 0, so F = 0 and T^{1} = T.

2. This establishes that T_{0} has the unique undercurrent T. The set E of 11.1 has T = ∂[E] + [E_{0}], T_{t} = ∂[E_{t}], and {χ_{π_{#}(E_{t})}}_{t≥0} weakly continuous in BV (R^{n}). Since {u > 0} ⊆ E ⊆ {u ≥ 0} and L^{n+1}({u = 0}) = 0, we can take E to be the open set {u > 0}.

3. By 9.2 and the uniqueness of T, T is a matching motion, that is, {μ_{T_{t}}}_{t≥0} is a Brakke motion. Q.E.D.

§12 Almost Everywhere Regularity

Let T be a boundary motion and μ_{t} = μ_{T_{t}}. Because {μ_{t}}_{t≥0} has unit density, Brakke’s Main Regularity Theorem [B, 6.12] immediately applies to prove a certain kind of partial regularity. Because T is locally a reduced boundary, we will use Brakke’s Local Regularity Theorem to prove a slightly different result, namely

H^{n}(sing (spt T)) = 0.

This is Theorem 12.9. Corollary 12.11 treats almost-everywhere regularity of almost every level-set of the level-set flow.

Let ω_{k} be the volume of the unit ball in R^{k}.

12.1 Brakke’s Local Regularity Theorem. ([B, 6.11]) There are constants 0 < c_{1} < c_{2} and η_{0} > 0, depending on n and k, such that if {μ_{t}}_{t≥0} is a unit density Brakke motion, that is,

Θ^{k}(μ_{t}, ·) = 1 μ_{t}-a.e. for all t ≥ 0,

and if

(i) R > 0, (x_{0}, t_{0}) ∈ R^{n} × [0, ∞) and P ⊆ R^{n} is a k-plane containing x_{0},
(ii) ∫_{B_{R}(x_{0})} |dist(x, P)|^{2} dμ_{t_{0}} ≤ η_{0}R^{k+2},
(iii) \frac{ω_{k}}{2} (\frac{R}{2})^{k} ≤ μ_{t}(B_{R/2}(x_{0})) < \frac{3ω_{k}}{2} (\frac{R}{2})^{k} for t ∈ [t_{0}, t_{0} + c_{2}R^{2}],
(iv) μ_{t}(B_{R}(x_{0})) < \frac{3}{2}ω_{k}R^{k} for t ∈ [t_{0}, t_{0} + c_{2}R^{2}],

then the intersection of

⋃_{t≥0} spt μ_{t} × {t}

with

B_{R/2}(x_{0}) × [t_{0} + c_{1}R^{2}, t_{0} + c_{2}R^{2}]

is a C^{∞} manifold.

We will need the following specialization of Brakke’s Clearing-Out Lemma.

12.2 Clearing-Out Lemma. ([B, 6.3]) There are constants η_{1} > 0, 0 < c_{3} < c_{4} depending on n and k such that for any integral Brakke motion {μ_{t}}_{t≥0}, R > 0, and (x_{0}, t_{0}) ∈ R^{n} × [0, ∞), if

μ_{t_{0}}(B_{R}(x_{0})) ≤ η_{1}R^{k}

then

μ_{t}(B_{R/2}(x_{0})) = 0 for t ∈ [t_{0} + c_{3}R^{2}, t_{0} + c_{4}R^{2}].

ELLIPTIC REGULARIZATION 69

Recall that the reduced boundary ∂^{∗}E of a set E of finite perimeter is defined to be the set of points x ∈ ∂E such that

(i) P = T_{x}(H^{n−1}⌊∂E) exists and is of multiplicity one.

(ii) lim_{λ↓0} χ_{E}(x + λy) = χ_{P^{+}}(y) in L^{1}, where P^{+} is a half-space bounded by P.

According to DeGiorgi’s theorem (see [S, 14.3]), ∂^{∗}E is rectifiable and ∂[E] = τ(∂^{∗}E, 1, ξ) where ξ orients T_{x}(H^{n−1}⌊∂E) as the boundary of P^{+}(x). In particular, μ_{∂[E]} = H^{n−1}⌊∂^{∗}E has unit density almost everywhere.

Let T be a boundary motion. Using the Clearing-Out Lemma, we will show that spt T is rectifiable and in fact μ_{T} = H^{n}⌊spt T. Then we will use the Local Regularity Theorem to show that for μ_{T} -a.e. (x, t), the existence of the space-time tangent plane gives smoothness near (x, t).

For small r, a cylinder B_{r}(x) × [t − r, t + r] has an intrinsic time scale much larger than the spatial radius. We will need to bound ∫_{t−r}^{t+r} ∫_{B_{r}} |H⃗| dμ_{t} dt over such cylinders in order to control the mass growth. If we knew in advance that the motion were smooth almost everywhere, then since |ẋ| = |H⃗| we would have

|Dτ| = \frac{1}{\sqrt{H^{2} + 1}},

where τ = time, D = D^{T}, and then

(†)  ∫_{t−r}^{t+r} ∫_{B_{r}} |H⃗| dμ_{t} dt = ∫_{B_{r}×[t−r,t+r]} |H⃗||Dτ| dμ_{T} ≤ μ_{T} (B_{r} × [t − r, t + r])

so that ∫∫ |H⃗| dμ_{t} dt is estimated in terms of the local mass of T. However, in the case at hand, we must use the Lebesgue-Besicovitch differentiation theorem and the global bound ∫∫ |H⃗| dμ_{t} dt < ∞.

By inserting ϕ = 1 in 6.3 and using Hölder’s inequality, we find

∫_{0}^{L} ∫ |H⃗| dμ_{t} dt < ∞,

where L is the extinction time. Define Radon measures σ, ν by

σ(ψ) = ∫∫ ψ dμ_{t} dt, ν(ψ) = ∫∫ ψ|H⃗| dμ_{t} dt,

70 TOM ILMANEN

for ψ ∈ C_{c}^{0}(R^{n} × [0, ∞), R^{+}). Evidently ν is absolutely continuous with respect to σ. Note also that by the support lemma 11.6(ii), σ and μ_{T} have the same null-sets. By DeGiorgi’s theorem, μ_{T} = H^{n}⌊X, where X is the reduced boundary of E relative to R^{n} × [0, ∞), and σ = σ⌊X, ν = ν⌊X. We obtain the following lemma by the Lebesgue-Besicovitch differentiation theorem [S, 4.7].

12.3. First H Lemma. Let T be a boundary motion with μ_{T} = H^{n}⌊X. Then for H^{n}-a.e. (x, t) ∈ X,

\frac{dν}{dσ} (x, t) ≡ lim_{r↓0} \frac{∫∫_{B_{r}^{n+1}(x,t)} |H⃗| dμ_{s}ds}{∫∫_{B_{r}^{n+1}(x,t)} dμ_{s}ds} = |H⃗(x, t)|.

For points not covered by the above statement, we have

12.4 Second H Lemma. Let T be a boundary motion with μ_{T} = H^{n}⌊X. Then for H^{n}-a.e. (x, t) ∈ R^{n} × [0, ∞) \ X,

Θ^{n}(ν, (x, t)) ≡ lim_{r↓0} \frac{1}{r^{n}} ∫_{t−r}^{t+r} ∫_{B_{r}^{n}(x)} |H⃗| dμ_{s}ds = 0.

Proof. According to [S, 3.5], if ν is a Borel-regular measure and X ⊆ R^{n} is νmeasurable with ν(X) < ∞, then Θ^{k}(ν⌊X, x) = 0 for H^{k}-a.e. x ∈ R^{n} \ X.

Replace k by n, n by n + 1. As noted above, ν = ν⌊X. Evidently X is ν-measurable and ν(X) < ∞. Thus Θ^{n}(ν, (x, t)) = 0 for H^{n}-a.e. x ∈ R^{n} × [0, ∞) \ X. This is the lemma. Q.E.D.

Let us henceforth write B_{r}(x) for B_{r}^{n}(x).

12.5 Long-Time Clearing-Out Lemma. There exists η_{2} > 0, r_{0} > 0 depending on n, k such that for any boundary motion T, (x_{0}, t_{0}) ∈ R^{n} × [0, ∞), and r ∈ (0, r_{0}], if

\begin{matrix}(i) & ∫_{t_{0}−r}^{t_{0}+r} ∫_{B_{r}(x_{0})} |H⃗| dμ_{t}dt ≤ η_{2}r^{n} \\ (ii) & ∫_{t_{0}−r}^{t_{0}−r/2} μ_{t}(B_{r}(x_{0})) dt ≤ η_{2}r^{n} \end{matrix}

then

μ_{T} (B_{r/4}(x_{0}) × [t_{0} − r/4, t_{0} + r]) = 0.

ELLIPTIC REGULARIZATION 71

12.6 Remarks. (a) Note that the time interval [t_{0} − r, t_{0} + r] is relatively long. Actually, any longer time interval works also.

(b) The quantity in (i) is the volume crossed by T_{t} — once we have established a certain amount of regularity. Thus to reach the center of a ball, a surface must either have a large area (on average), cross a large volume, or do it right away.

(c) Once we have proven almost everywhere regularity of spt T, we can use (†) and the proof of 12.7 to replace hypotheses (i) and (ii) by the single hypothesis

μ_{T} (B_{r}(x_{0}) × [t_{0} − r, t_{0} + r]) ≤ η_{2}r^{n}.
Proof of 12.5. By (ii), there exists t_{1} ∈ [t_{0} − r, t_{0} − r/2] with
μ_{t_{1}}(B_{r}) ≤ 2η_{2}r^{n−1}.

Choose ϕ ∈ C_{c}^{2}(B_{r}(x_{0}), R^{+}) with ϕ = 1 on B_{r/2}(x_{0}), 0 ≤ ϕ ≤ 1, |∇ϕ| ≤ 3/r. Then for t ∈ [t_{0} − r/2, t_{0} + r],

μ_{t}(B_{r/2}(x_{0})) ≤ μ_{t}(ϕ)
≤ μ_{t_{1}}(ϕ) + ∫_{t_{1}}^{t_{0}+r} ∫_{B_{r}(x_{0})} −ϕH^{2} + ∇ϕ · S^{⊥} · H⃗ dμ_{t}dt
≤ 2η_{2}r^{n−1} + max |∇ϕ| ∫_{t_{0}−r}^{t_{0}+r} ∫_{B_{r}(x_{0})} |H⃗| dμ_{t}dt
≤ 5η_{2}r^{n−1} by (i).

Now choose η_{2} according to

5η_{2} = (\frac{1}{2})^{n−1} ω_{n−1}η_{1}.

Then by the Clearing-Out Lemma 12.2,

μ_{t}(B_{r/4}(x_{0})) = 0 for t ∈ [t_{0} − r/2 + c_{3}r^{2}/4, t_{0} + r].

Choose r_{0} so that c_{3}r^{2}/4 ≤ r/4 for r ≤ r_{0}. Then

μ_{t}(B_{r/4}(x_{0})) = 0 for t ∈ [t_{0} − r/4, t_{0} + r],

from which the lemma follows by the Support Lemma 11.6(iii). Q.E.D.

72 TOM ILMANEN

12.7 Rectifiability of Support. Let T be a boundary motion. Then spt T is rectifiable and

μ_{T} = H^{n}⌊spt T.

Evans and Spruck have proven a related result in [ES3], namely rectifiability of the boundary of the level-set flow for a.e. t ≥ 0.

Proof. Define X by

X = {(x, t) : Θ^{n}(μ_{T} , (x, t)) = 1}.

Then μ_{T} = H^{n}⌊X by Lemma 1.3 and DeGiorgi’s theorem on sets of finite perimeter. Define

X_{∗} = {(x, t) : Θ^{n}(μ_{T} , (x, t)) does not exist, or lies in (0, 1) ∪ (1, ∞)},
X_{0} = {(x, t) ∈ spt T : Θ^{n}(μ_{T} , (x, t)) = 0}.

Then spt T = X ∪ X_{∗} ∪ X_{0}. By 1.3, H^{n}(X_{∗}) = 0. For any fixed (x, t) ∈ X_{0}, there exists r_{2} ∈ (0, r_{0}] such that for all r ≤ r_{2},

∫_{t−r}^{t+r} μ_{t}(B_{r}(x)) dt = ∫_{B_{r}(x)×[t−r,t+r]} |Dτ|dμ_{T}  by the Slice Lemma
≤ μ_{T} (B_{r}(x) × [t − r, t + r])
≤ η_{2}r^{n}.

By Lemma 12.4, for H^{n}-a.e. (x, t) ∈ X_{0} we can further decrease r so that in addition

∫_{t−r}^{t+r} ∫_{B_{r}(x)} |H⃗| dμ_{s}ds ≤ η_{2}r^{n}.

Then by the Long-Term Clearing-Out Lemma 12.5, (x, t) ∉ spt T. Since X_{0} ⊆ spt T, this shows H^{n}(X_{0}) = 0. Thus μ_{T} = H^{n}⌊X = H^{n}⌊spt T. Q.E.D.

Remark. Once we have proven regularity almost everywhere of spt T, we see that actually X_{0} = ∅, by remark 12.6(c).

Let P be an (n − 1)-plane in R^{n}, and let P^{+}, P^{−} be the two components of R^{n} \ P. Define

P_{a} = {x ∈ R^{n} : dist(x, P) ≤ a},  P_{a}^{+} = P^{+} \ P_{a},  P_{a}^{−} = P^{−} \ P_{a}.

ELLIPTIC REGULARIZATION 73

We turn our attention to a parabolic cylinder where spt T remains close to P × time and ∫∫ |H| dμ_{t} dt is small. (Later these conditions can be achieved by blowing up parabolically by some finite amount.)

12.8 Local Regularity of T. There exist r_{1} > 0, δ_{0} > 0 depending only on n, such that for any boundary motion T, (x_{0}, t_{0}) ∈ R^{n} × (0, ∞), r ∈ (0, min(r_{1}, t_{0})], and plane P ∋ x_{0}, if

(i)  ∫_{t_{0}−r}^{t_{0}−r/2} μ_{t}(B_{r}(x_{0})) dt ≤ (1 + δ_{0})ω_{n−1} (\frac{r}{2}) r^{n−1}
(ii)  ∫_{t_{0}−r}^{t_{0}−r/2} μ_{t}(B_{r/2}(x_{0})) dt ≤ (1 + δ_{0})ω_{n−1} (\frac{r}{2})^{n}
(iii)  ∫_{t_{0}−r}^{t_{0}−r/2} μ_{t}(B_{r}(x_{0}) \ P_{δ_{0}r}) dt ≤ δ_{0}r^{n}
(iv)  ∫_{t_{0}−r}^{t_{0}+r} ∫_{B_{r}(x_{0})} |H⃗| dμ_{t}dt ≤ δ_{0}r^{n}
(v)  L^{n+1}(E ∩ F^{+}) ≥ \frac{7}{8}L^{n+1}(F^{+})
(vi)  L^{n+1}(E ∩ F^{−}) ≤ \frac{1}{8}L^{n+1}(F^{−}),

where F^{±} = (B_{r}(x_{0}) ∩ P_{δ_{0}r}^{±}) × [t_{0} − r, t_{0} + r], then

spt T ⋂(B_{r/4}(x_{0}) × [t_{0} − r/4, t_{0} + r])

is a C^{∞} manifold.

= 0.880cm  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} F^{−}  P  F^{+}  t_{0} − r t_{0} + r t_{0} P_{a}^{−}  P_{a}^{+} P_{a}

Figure 12. Regions.

Proof. The proof is a straightforward application of Brakke’s Local Regularity Theorem and the Long-Term Clearing-Out Lemma.

1. Let us first specify the constants. Define r_{1} = 1/(2c_{1}). Next, choose ε > 0 so that

(1)  \begin{matrix}\frac{1}{(1 − ε)^{n−1}} < \frac{3}{2},  c_{3}ε^{2}r_{1} < 1, & \frac{6ε^{2}}{(1 − ε)^{2}} ω_{n−1} ≤ η_{0}, \\ (\frac{3ε}{1 − ε})^{2} + (\frac{1 − \sqrt{ε}}{2})^{2} ≤ (\frac{1}{2})^{2} ,  (1− & \sqrt{ε})^{n−1} > \frac{1}{2},  ε < \frac{1}{10}. \end{matrix}

Next, take δ_{0} ≡ δ < ε so small that

(2) \frac{1}{(1 − ε)^{n−1}} ((1 + δ)ω_{n−1} + \frac{2^{n+1}δ}{ε} ) < \frac{3}{2}ω_{n−1},  δ (4 + \frac{3}{ε}) ≤ η_{1} (\frac{ε}{2})^{n−2} .

Define R = (1 − ε)r. We will refer to (1) and (2) as needed.

2. Assume for convenience that (x_{0}, t_{0}) = (0, 0) (with starting time before zero). Our aim is to apply Brakke’s Local Regularity Theorem to B_{R}. Let us work on μ_{t}(B_{R}) first.

By (i), there is t_{1} ∈ [−r, −r/2] such that
μ_{t_{1}}(B_{r}) ≤ (1 + δ)ω_{n−1}r^{n−1}.

Let ϕ ∈ C_{c}^{2}(B_{r}, R^{+}) be a function such that ϕ ≡ 1 on B_{R}, 0 ≤ ϕ ≤ 1 and |∇ϕ| ≤ 2/(εr). Then for all t ∈ [−r/2, r],

(3)

μ_{t}(B_{R}) ≤ μ_{t}(ϕ)

≤ (1 + δ)ω_{n−1}r^{n−1} + max |∇ϕ| ∫_{−r}^{r} ∫_{B_{r}} |H⃗| dμ_{t}dt

≤ (1 + δ)ω_{n−1}r^{n−1} + \frac{2}{εr} δr^{n} by (iv)

≤ \frac{(1 + δ)ω_{n−1} + 2δ/ε}{(1 − ε)^{n−1}}R^{n−1}

< \frac{3}{2}ω_{n−1}R^{n−1} by (2).

3. Similarly, starting from (ii) we obtain for all t ∈ [−r/2, r],

(4)  \begin{aligned}μ_{t}(B_{R/2}) & ≤ \frac{(1 + δ)ω_{n−1} + 2^{n+1}δ/ε}{(1 − ε)^{n−1}} (\frac{R}{2})^{n−1} \\ & < \frac{3}{2}ω_{n−1} (\frac{R}{2})^{n−1} . \end{aligned}

4. Our next goal is to clear out the region not near P. Let B_{εr}(x) ⊆ B_{r} be any ball disjoint from P_{δr}. By (iii), there is t_{1} ∈ [−r, −3r/4] with

μ_{t_{1}}(B_{εr}) ≤ 4δr^{n−1}.

As above, we find for t ∈ [−3r/4, r],

μ_{t}(B_{εr/2}) ≤ 4δr^{n−1} + max |∇ϕ| ∫_{−r}^{r} ∫_{B_{r}} |H⃗| dμ_{t}dt

≤ 4δr^{n−1} + \frac{3}{εr} δr^{n} by (iv)

≤ η_{2} (\frac{εr}{2} )^{n−1} by choice of δ in (2).

By the Clearing-Out Lemma 12.2, we obtain

μ_{t}(B_{εr/4}(x)) = 0 for t ∈ [−3r/4 + c_{3}ε^{2}r^{2}/4, r].

Considering all such balls, since R = (1 − ε)r, δ < ε, r ≤ r_{1}, and c_{3}ε^{2}r_{1}/4 < 1/4 by (1), we obtain

(5)  μ_{t}(B_{R} \ P_{2εr}) = 0  for t ∈ [−r/2, r].

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^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} B_{r}  B_{R} B_{εr}

Figure 13. Picture of B_{R} \ P_{2εr}.
5. Then for t ∈ [−r/2, r],

(6)

∫_{B_{R}} |dist(x, P)|^{2}dμ_{t} ≤ (2εr)^{2}μ_{t}(B_{R})  by (5)

= \frac{6ε^{2}}{(1 − ε)^{2}} ω_{n−1}R^{n+1} by (3)

≤ η_{0}R^{n+1} by choice of ε in (2).

76 TOM ILMANEN

6. Consider the open (n − 1)-disk, and cylinder

D ≡ int B_{(1−\sqrt{ε})R/2}^{n−1}(0) ⊆ P, C ≡ D × [−3εr, 3εr] ⊆ R^{n}.

Note that the condition

(\frac{R(1 − \sqrt{ε})}{2})^{2} + (\frac{3εR}{1 − ε})^{2} ≤ (\frac{R}{2})^{2}

from (1) guarantees that C ⊆ B_{R/2}.

= 0.880cm  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B_{R/2} T_{t} D C 0 1

Figure 14. Picture of T_{t}⌊C.

It follows from (5) by 11.6(iii) and μ_{T_{t}} = μ_{t} that

(7) spt T_{t} ∩ B_{R} ⊆ P_{2εr}, spt ∂[E] ∩ (B_{R} × [−r/2, r]) ⊆ P_{2εr} × [−r/2, r].

Hence spt ∂(T_{t}⌊C) ⊆ ∂D × [−2εr, 2εr], so letting π : \overline{C} → \overline{D} be the projection map,

∂(π_{#}(T_{t}⌊C))⌊D = 0.

By the Constancy Theorem [S, 26.27], π_{#}(T_{t}⌊C) is a constant integer multiple of [D]. By (7), (v), (vi) and the Constancy Theorem

E_{t} ⊇ B_{R} ∩ P_{2εr}^{+}, E_{t} ∩ (B_{R} ∩ P_{2εr}^{−}) = ∅.

Therefore the Constancy Theorem, the multiple cannot be zero, and we obtain a lower bound for the mass in B_{R/2},

(8)

μ_{t}(B_{R/2}) ≥ M[T_{t}⌊C]

≥ M[π_{#}(T_{t}⌊C)]

≥ M [[D]] by the above

= (1 − \sqrt{ε})^{n−1}ω_{n−1} (\frac{R}{2})^{n−1}

≥ \frac{1}{2}ω_{n−1} (\frac{R}{2})^{n−1} by (1)

for all t ∈ [−r/2, r].

8. We now apply Brakke’s Local Regularity Theorem 12.1, using (3), (4), (6), (8), to conclude that

(⋃ spt μ_{t} × \{t\}) ⋂ (B_{R/2} × [−\frac{r}{2} + c_{1}R^{2}, r])

is a smooth manifold. Note that c_{1}R^{2} ≤ c_{1}r^{2} ≤ r/2 by virtue of r ≤ r_{1}, and r/2 < R. Using 11.6(iii) we find that

spt T ⋂ (B_{r/4} × [−\frac{r}{4}, r])

is a smooth manifold. Q.E.D.

Now we present the main results of this section.

12.9 Almost Everywhere Regularity Theorem. Let T be a boundary motion. Then H^{n}(sing(spt T)) = 0.

Together with 11.4 this implies

12.10 Almost Everywhere Regularity for Nonfattening Level-Set. If \{Γ_{t}\}_{t≥0} is a level-set motion that does not fatten up, then \overline{∂^{∗}\{u > 0\}} is smooth H^{n}- almost everywhere.

However, it has not been proven that {u = 0} is the closure of the reduced boundary of {u > 0} or even its topological boundary. This seems tricky; see Appendix. On the other hand, the theory of slicing and the Clearing-Out Lemma of [ES3] for level-sets allows us to prove the following. We write Z_{+} for Z ∩ (R^{n} × [0, ∞)).

78 TOM ILMANEN

12.11 Almost Everywhere Regularity of Almost Every Level-Set. Let f be a Lipschitz function on R^{n} with compact level-sets, and let u : R^{n} × [0, ∞) → R be the corresponding solution of equation §10(⋆). Then for a.e. a ∈ R,

(i) \{u = a\}_{+} = (\overline{∂^{∗}\{u > a\}})_{+} and \{u = a\}_{+} = (∂^{∗}\{u > a\})_{+} H^{n}-a.e.

(ii) H^{n}(sing({u = a})) = 0.

(iii) For a.e. t ≥ 0, H^{n−1}(sing(Γ_{t}^{a})) = 0.

Proof of 12.9. 1. Let X be the set of points (x, t) ∈ spt T such that

(a) (x, t) ∈ ∂^{∗}E.

(b) T_{(x,t)}T is not parallel to R^{n} × {0}.

(c) These two quantities exist and are equal:

lim_{r↓0} \frac{∫∫_{B_{r}^{n+1}(x,t)} |H⃗| dμ_{t′}dt′}{∫∫_{B_{r}^{n+1}(x,t)} dμ_{t′}dt′} = |H⃗(x, t)|.

By DeGiorgi’s theorem and the hypothesis that T is a boundary motion, we see that (a) holds μ_{T} -a.e. By 11.6(i), statement (b) holds μ_{T} -a.e. Lemma 12.3 says that (c) holds μ_{T} -a.e. We have seen in the rectifiability lemma 12.7 that μ_{T} = H^{n}⌊spt T. Hence

X = spt T H^{n}-a.e.

Therefore it suffices to show that for H^{n}-a.e. (x, t) ∈ X, there is a neighborhood U ∋ (x, t) such that spt T ∩ U is smooth.

2. Define

X_{A} = \{(x, t) ∈ X : |Dτ(x, t)| ≥ \frac{1}{A} and |H⃗(x, t)| ≤ A\},

where D = D^{T}, τ = time. Then X = ⋃_{A>0} X_{A}. Define the parabolic blowup

ρ_{λ}(x, t) = (\frac{x}{λ}, \frac{t}{λ^{2}} ) = (y, s),  T^{λ} = (ρ_{λ})_{#}(T),  X^{λ} = ρ_{λ}(X).

Then T^{λ} is also a boundary motion.

Let us check how properties (a)-(c) transform under ρ_{λ}. For all (x, t) ∈ X_{A},

|H⃗_{T_{s}^{λ}}(y, s)| = λ|H⃗_{T_{t}}(x, t)| ≤ λA.

Applying the Lebesgue-Besicovitch differentiation theorem to X^{λ} (as in 12.3), for μ_{T} -a.e. (x, t) ∈ X_{A},

(1)  lim_{r↓0} \frac{∫∫_{B_{r}^{n+1}(y,s)} |H⃗_{T_{s′}^{λ}} | dμ_{s′}ds′}{∫∫_{B_{r}^{n+1}(y,s)} dμ_{s′}ds′} = |H⃗_{T_{s}^{λ}}(y, s)| ≤ λA.

Conceivably T_{(y,s)}T^{λ} does not exist, even where T_{(x,s)}T does exist. But by the rectifiability of T, for μ_{T} -a.e. (x, t) ∈ X_{A}, T_{(y,s)}T^{λ} does exist and equals (ρ_{λ})_{#}(T_{(x,t)}T). Then for μ_{T} -a.e. (x, t) ∈ X_{A},

(2)  \begin{aligned}|D^{T^{λ}}τ(y, s)| & = \frac{|D^{T} τ(x, t)|}{\sqrt{|D^{T} τ(x, t)|^{2} + λ^{2}(1 − |D^{T} τ(x, t)|^{2})}} \\ & ≥ \frac{1}{\sqrt{1 + λ^{2}(A^{2} − 1)}}. \end{aligned}

Let δ_{0} be as in 12.8. If we select λ_{0} = λ_{0}(A, δ_{0}) sufficiently small, then (1) and (2) imply that for λ ≤ λ_{0} and μ_{T} -a.e. (x, t) ∈ X_{A},

(3)  |D^{T^{λ}}τ(y, s)| ≥ 1 − \frac{δ_{0}^{2}}{4} ,

(4)  lim_{r↓0} \frac{∫∫_{B_{r}^{n+1}(y,s)} |H⃗_{T_{s′}^{λ}} | dμ_{s′}ds′}{∫∫_{B_{r}^{n+1}(y,s)} dμ_{s′}ds′} ≤ \frac{δ_{0}}{4(\sqrt{2})^{n}ω_{n}} .

Furthermore, for μ_{T} -a.e. (x, t) ∈ X_{A}, properties (a), (b), and (c) remain valid for T^{λ} at (y, s).

3. We wish to apply the Local Regularity Lemma 12.8 at a point (y, s) satisfying the properties above. For simplicity assume (y, s) = (0, 0) (with negative starting time). Define Q = T_{(0,0)}T^{λ}, P = Q ∩ (R^{n} × {0}). Inequality (3) implies that

(5)  Q ∩ (B_{1} × [−1, 1]) ⊆ P_{δ_{0}} × [−1, 1]

= 0.880cm  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} P Q 0  1  t = 0

Figure 15. Parabolically Blown-up Tangent Plane.

According to (a) for T^{λ} at (0,0), as r ↓ 0 the homothetic blowup (μ_{T^{λ}})_{(0,0),r} converges as a Radon measure to H^{n}⌊Q. Let ψ be any test function in C_{c}^{0}(B_{1} ×[−1, 1], R^{+}). For small enough r, we have

∫∫ \frac{1}{r^{n}} ψ (\frac{y}{r} , \frac{s}{r}) dμ_{s}^{λ}(y) ds = ∫ \frac{1}{r^{n}} ψ (\frac{y}{r} , \frac{s}{r}) |D^{T^{λ}}τ(y, s)| dμ_{T} (y, s) by co-area

≤ ∫ \frac{1}{r^{n}} ψ (\frac{y}{r} , \frac{s}{r}) dμ_{T} (y, s)

≈ ∫_{Q} ψ dH^{n}

= \frac{1}{|D^{Q}τ|} ∫ ∫_{Qs̃} ψ dH^{n−1}ds̃

where Qs̃ = Q ∩ (R^{n} × {s̃})

≤ \frac{1}{1 − δ_{0}^{2}/4} ∫ ∫_{Qs̃} ψ dH^{n−1}ds̃ by (3)

≤ (1 + \frac{δ_{0}^{2}}{2} ) ∫ ∫_{Qs̃} ψ dH^{n−1}ds̃.

Take ψ ∈ C_{c}^{0}(B_{2} × [−2, 2]) to approximate the characteristic function χ_{B_{1}×[−1,−1/2]} from above. Then by the preceding,

\frac{1}{r^{n}} ∫_{−r}^{−r/2} μ_{s}^{λ}(B_{r}) ds ≤ \frac{1}{r^{n}} ∫∫ ψ (\frac{y}{r} , \frac{s}{r}) dμ_{s}^{λ}(y) ds

≤ (1 + \frac{δ_{0}^{2}}{2} ) ∫∫ ψ dH^{n−1}ds + o(1) (as r ↓ 0)

≤ \frac{1}{2}(1 + δ_{0})ω_{n−1},

ELLIPTIC REGULARIZATION 81

for a good enough approximation and small enough r. This is hypothesis 12.8(i) of the Local Regularity Theorem. Similarly, by taking

ψ ≈ χ_{B_{1/2}×[−1,−1/2]}, ψ ≈ χ_{(B_{1}\P_{δ_{0}} )×[−1,1]},

we obtain hypotheses 12.8(ii), (iii), for small enough r. Also, (5) above, together with statement (b) for T^{λ}, implies 12.8(v), (vi).

4. Now let us work on the remaining hypothesis, 12.8(iv). Using (4), we have

∫_{−r}^{r} ∫_{B_{r}} |H⃗| dμ_{s}^{λ}ds ≤ \frac{δ_{0}}{4(\sqrt{2})^{n}ω_{n}} ∫∫_{B_{\sqrt{2}r}^{n+1}(0,0)} dμ_{s}^{λ}ds  by (4)

≤ \frac{δ_{0}}{4(\sqrt{2})^{n}ω_{n}} μ_{T^{λ}}(B_{\sqrt{2}r}^{n+1}(0, 0)) by co-area

≤ \frac{δ_{0}}{4(\sqrt{2})^{n}ω_{n}} 2ω_{n}(\sqrt{2}r)^{n} by (3)

= δ_{0}r^{n},

which is hypothesis 12.8(iv).

5. It follows that (0,0) is a smooth point. Since (0,0) represented ρ_{λ}(x, t) for μ_{T} -a.e. (x, t) ∈ X_{A} and X = ⋃_{A>0} X_{A}, it follows that μ_{T} -a.e. (x, t) ∈ X is smooth. Therefore H^{n}-a.e. (x, t) ∈ spt T is smooth, proving 12.9. Q.E.D.

Proof of 12.11.

(i) Note that ∂^{∗}{u > a}_{+} ⊆ ∂{u > a}_{+} ⊆ {u = a}. According to [ES1], Lip u(·, ·) ≤ Lip u(·, 0) < ∞. Applying 1.21 with U = R^{n} × (0, ∞) we obtain for a.e. a ∈ R,

τ({u = a}_{+}, 1, ξ_{a}) = τ((∂^{∗}{u > a})_{+}, 1, ξ_{∂^{∗}{u>a}_{+}}).

It follows that for a.e. a ∈ R,

(1)  H^{n}({u = a}_{+} \ ∂^{∗}{u > a}_{+}) = 0.

Next we show

\{u = a\}_{+} = (\overline{∂^{∗}\{u > a\}})_{+}.

Suppose (x, t) ∈ {u = a}_{+}, so x ∈ Γ_{t}^{a}, t > 0. According to the Clearing-Out Lemma of Evans-Spruck [ES3, 3.1] for the level-set flow, there exist constants β > α > 0, η > 0 such that for all r > 0,

H^{n−1}(Γ_{s}^{a} ∩ B_{r}(x)) > ηr^{n−1} for s ∈ [t − βr^{2}, t − αr^{2}] ∩ [0, ∞),

82 TOM ILMANEN where B_{r} = B_{r}^{n}. Then, provided 0 < r^{2} ≤ t/β,

H^{n}({u = a} ∩ (B_{r}^{n+1}(x) × [t − βr^{2}, t − αr^{2}])) ≥ ∫_{t−βr^{2}}^{t−αr^{2}} H^{n−1}(Γ_{s}^{a} ∩ B_{r}(x)) ds
≥ ηr^{n−1}(β − α)r^{2}

from which it follows by (1) that

∂^{∗}{u > a} ∩ (B_{r}(x) × [t − βr^{2}, t − αr^{2}]) ≠ ∅.

for arbitrarily small r. We conclude

(x, t) ∈ (\overline{∂^{∗}\{u > a\}})_{+}

for (x, t) ∈ \{u = a\}_{+}. Along with the first remark this shows (\overline{∂^{∗}\{u > a\}})_{+} = \{u = a\}_{+}.

(ii) For all but countably many level-sets, L^{n+1}({u = a}) = H^{n}({f = a}) = 0. Therefore (ii) follows immediately from (i) and 12.10.

(iii) follows trivially from (ii). Q.E.D.

Appendix

Here are some open problems. I make no claim to originality here; it would not even be possible to formulate these questions without the foundational work of Federer, Almgren, Allard, Brakke, Simon, as well as Gage, Hamilton, Grayson, Huisken, Angenent and others. These problems are quite hard, and serve as distant, blinding lights.

Brakke Motion and Level-Set Flow

A. Better Partial Regularity. Is it possible to improve the result of §12 somewhat, say to H^{n−1}(sing(spt T)) = 0 for all t ≥ 0?

B. Motion of Edge. (Evans-Spruck) If a level-set fattens, is there a unit density boundary motion on ∂{u(·, t) > 0}?

C. Equality in Brakke’s Inequality. For the unit density boundary motion of §11, show that Brakke’s inequality is actually an equality (for almost every time).

D. No Mass Drop. For the unit density boundary motion of §11, show M[μ_{t}] is continuous in time. (This would imply H^{n−1}-a.e. regularity for all times; see the final theorem of [B].)

E. Unit Density. (A very hard and old problem.) Can we show unit density for the motion of boundaries of sets? I.e.: If (T, {μ_{t}}_{t≥0}) is an enhanced pair, with T_{0} the boundary of an open set, is T a boundary motion?

Originally I thought that unit density might be preserved for any Brakke motion. The following simple example, the sum of two homothetic spoons, shows that this is false:

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Figure 16. Double Density Develops.

(D) and (E) are related, because mass loss for the undercurrent should occur at the same time that two pieces of the varifold double up.

84 TOM ILMANEN

The following example suggests: For boundaries, multiple density can form in infinite time but not in finite time.

Example. (with Evans) There is a surface in R^{3} which looks like two parallel planes attached by a tube, that evolves to form a double density plane in infinite time.

F. Nonfattening. (Evans-Spruck, DeGiorgi [D]) If a level-set fattens, it fattens at t = 0. (In particular, a smooth set never fattens, even past singularities.)

G. Criterion for Nonfattening. When does a set X fatten? This is especially interesting if X is the support of a stationary varifold.

Example. ([I5]) A minimizing current (of finite mass) never fattens. On the other hand, a stationary, non-minimizing cone fattens, at least if sing(spt T_{0}) = {0}.

Role of minimization.

Minimizing currents are much smoother that merely stationary ones. Brakke’s regularity theory is of the “stationary” character, that is, it only uses the Euler-Lagrange equation. Normally nothing further is possible for parabolic equations, since

(a) There is no minimization principle.

(b) Stationary objects for the elliptic equation solve the parabolic equation. On the other hand, in certain low-dimensional examples of nonunique evolution,

some of the flows decrease surface area faster and are more regular. These seem to be the ones that arise as limits of the minimizers of the elliptic regularization scheme.

The simplest example of this phenomenon is the cross in R^{2} (see the figure in 6.5). More generally, the boundary of the fattening stationary cone of [I5] is smooth after time t = 0.

H. Characterization of Limit. Characterize these limits of minimizers intrinsically.

I. Higher Regularity. Show that limits of minimizers for the mean curvature flow have very high regularity, analogous to the codimension seven regularity for the elliptic minimizers.

One can ask the same question for the time-step minimization method introduced in Luckhaus [Lu], Almgren-Wang [AW] and Almgren-Taylor-Wang [ATW].

ELLIPTIC REGULARIZATION 85

Blowups.

It is possible to pass back and forth between translative and homothetic solitons by means of blowup and blowdown, along the lines of the elliptic regularization construction.

J. Correspondence. For what class of homothetic and translative solitons is this a one-to-one correspondence?

It is possible to blow up any Brakke motion parabolically at any point by Huisken’s method to obtain a (possibly nonsmooth) homothetic soliton. If the Brakke motion was the limit of minimizers, the parabolic blowup can be obtained directly, via a diagonal subsequence, as the blowdown of a sequence of minimizing translative solitons P^{i} (with boundary).

K. Provenance of the Blowup. Can the parabolic blowup of a Brakke motion that is the limit of minimizers also be obtained by blowing down a single complete, minimizing translative soliton?

If true, this would give a general tool for studying the asymptotic behavior of singularities.

Positive Mean Curvature.

We conjecture

L. Classification of Positive Translative Solitons. A complete, minimizing translative soliton T ⊆ R^{n+1} (n ≤ 7) of positive mean curvature is either

(a) Σ^{k} × R^{n−k−1} (possibly tilted) where Σ^{k} is a rotationally symmetric translative soliton in R^{k+1}, k ≥ 2, or

(b) Γ × R^{n−2} (possibly tilted), Γ = grim reaper.

If we allow nonnegative mean curvature, we also get planes.

M. Positive Curvature Blowups. Suppose the motion has positive mean curvature. Then only a blowdown of (a) above can occur as the blowup of a motion that is obtained as the limit of minimizers. That is, the only blowups of singularities are the sphere and cylinders.

Huisken has already proven this conjecture under the hypothesis that the limit is a smooth surface.

86 TOM ILMANEN

Generic Statements

N. Genericity of Nonnegative Mean Curvature. (Huisken) Show for all but countably many level sets, each blowup has curvature of one sign.

This can be proven by a topological argument under the strong additional hypothesis that each flow is smooth on a suitable small neighborhood leading up to the critical point.

With (M), this would show that for the mean curvature flow, most blowups are spheres or cylinders, up to n = 7. Neckpinches suggest that there will be a point singularity, but this is not quite true, as shown by the following example.

Marriage-Ring Example. A thin, rotationally symmetric torus in R^{3} will shrink to a circle of singularities.

O. Strongest Regularity Conjecture.

(a) For n ≤ 7, any boundary motion in R^{n} can be perturbed slightly to a motion with a finite number of singularities in space-time (past a given time).

(b) For n > 7, any boundary motion can be perturbed to a motion with codimension seven singularities except for a finite number of transition times (past a given time).

Dimension Three

P. Singularities. In R^{3}, show that any singularity of a boundary motion that arises a the limit of minimizers is either

(a) a point singularity, or

(b) topologically equivalent to the marriage-ring example.

This includes the conjecture that a singularity with cylinder as blowup is topologically a neckpinch, cusp, or peanut, unless it is the marriage-ring. The conjecture is due to Grayson, Hamilton, and Huisken. In particular, it would show that there are only a finite number of singularities (either a point or a marriage-ring) past a given time. The following speculation would strengthen this slightly.

Q. Finitude of Spheres. In R^{3}, a shrinking sphere occurs as the parabolic blowup at only a finite number of points.

Using such results, it would be possible to estimate the number of singularities in space-time using Euler characteristic. See Ecker [Ec], Toro [T] for interesting work.

References

[Al] Allard, W., “On the First Variation of a Varifold,” Annals of Mathematics 95 (1972) 417-491.

[AC] Allen, S., and Cahn, J., “A Microscopic Theory for Antiphase Boundary Motion and its Application to Antiphase Domain Coarsening,” Acta Metall. 27 (1979) 1084-1095.

[A] Almgren, F., “Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems among Surfaces of Varying Topological Type and Singularity Structure,” Annals of Math. 87 (1968) 321-391.

[AW] Almgren, F., and Wang, L., “Mathematical Existence of Crystal Growth with Gibbs-Thompson Curvature Effects,” in preparation.

[ATW] Almgren, F., Taylor, J., and Wang, L., “Curvature-Driven Flows: A Variational Approach,” preprint.

[AAG] Altschuler, S., Angenent, S., and Giga, Y., “Mean Curvature Flow Through Singularities for Surfaces of Rotation,” Hokkaido Univ. Preprint Series in Math. #130, Dec. 1991.

[An1] Angenent, S., “Parabolic Equations for Curves on Surfaces (I). Curves with p-Integrable Curvature,” Annals of Math. 132 (1990) 451-483.

[An2] Angenent, S., “Parabolic Equations for Curves on Surfaces (II). Intersections, Blow Up and Generalized Solutions,” Annals of Math. 133 (1991) 171-215.

[An3] Angenent, S., “On the Formation of Singularities in the Curve Shortening Flow,” J. Diff. Geom. 33 (1991) 601-633.

[An4] Angenent, S., “Shrinking Donuts,” Proc. of the Conf. on Elliptic and Parabolic Equations held at Gregynog, Wales, Aug. 1989.

[BSS] Barles, G., Soner, H. M., and Souganidis, P. E., “Front Propagation and Phase Field Theory,” SIAM J. Cont. Opt., to appear.

[B] Brakke, K., The Motion of a Surface by its Mean Curvature, Princeton Univ. Press, 1978.

[BK] Bronsard, L., and Kohn, R. V., “Motion by Mean Curvature as the Singular Limit of Ginzburg-Landau Dynamics,” J. Diff. Eq. 90 (1991) 211-237.

[Ch] Chen, Y., “The Weak Solutions to the Evolution Problem of Harmonic Maps,” Math. Z. 201 (1989) 69-74.

[CS] Chen, Y., and Struwe, M., “Existence and Partial Regularity Results for the Heat Flow for Harmonic Maps,” Math. Z. 201 (1989) 83-103.

[CGG] Chen, Y.-G., Giga, Y., and Goto, S., “Uniqueness and Existence of Viscosity Solutions of Generalized Mean Curvature Flow Equations,” J. Diff. Geom. 33 (1991) 749-786.

[CIL] Crandall, M. G., Ishii, H., and Lions, P.-L., “User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations,” Bull. of the Amer. Math. Soc. 27 (1992) 1-67.

[D] DeGiorgi, Ennio, “Some Conjectures on Flow by Mean Curvature,” white paper, 1990.

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[Ec] Ecker, K., “An ε-Regularity Theorem for Mean Curvature Flow of Surfaces in Three-Manifolds,” preprint.

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[EH2] Ecker, K., and Huisken, G., “Interior Estimates for Hypersurfaces Moving by Mean Curvature,” preprint, Centre for Math. Analysis, Austr. Nat. Univ., 1990.

[ES1] Evans, L. C., and Spruck, J., “Motion of Level Sets by Mean Curvature I,” J. Diff. Geom. 33 (1991) 635-681.

[ES2] Evans, L. C., and Spruck, J., “Motion of Level Sets by Mean Curvature II,” Trans. of the Amer. Math. Soc., to appear.

[ES3] Evans, L. C., and Spruck, J., “Motion of Level Sets by Mean Curvature III,” J. Geom. Analysis, to appear.

[ES4] Evans, L. C., and Spruck, J., “Motion of Level Sets by Mean Curvature IV,” preprint.

[ESS] Evans, L. C., Soner, H. M., and Souganidis, P. E., “Phase Transitions and Generalized Motion by Mean Curvature,” Comm. Pure Appl. Math., to appear.

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[Gr1] Grayson, M. A., “The Heat Equation Shrinks Embedded Curves to Round Points,” J. Diff. Geom. 26 (1987) 285-314.

ELLIPTIC REGULARIZATION 89

[Gr2] Grayson, M. A., “Shortening Embedded Curves,” Annals of Math. 129 (1989) 71-111.

[Ha1] Hamilton, R., “Three-Manifolds with Positive Ricci Curvature,” J. Diff. Geom. 17 (1982) 255-306.

[Ha2] Hamilton, R., “Four-Manifolds with Positive Curvature Operator,” Journal of Differential Geometry 24 (1986) 153-179.

[Ha3] Hamilton, R., “The Ricci Flow on Surfaces,” Contemporary Mathematics 71 (1988) 237-261.

[Ha4] Hamilton, R., “The Harnack Estimate for the Ricci Flow,” preprint.

[Ha5] Hamilton, R., “A Compactness Property for Solutions of the Ricci Flow,” preprint.

[Ha6] Hamilton, R., “Eternal Solutions to the Ricci Flow,” preprint.

[Ha7] Hamilton, R., “An Isoperimetric Estimate for the Ricci Flow on the Two-Sphere.”

[Ha8] Hamilton, R., “Monotonicity Formulas for Parabolic Flows on Manifolds,” preprint.

[H1] Huisken, G., “Flow by Mean Curvature of Convex Surfaces into Spheres,” J. Diff. Geom. 20 (1984) 237-266.

[H2] Huisken, G., “Asymptotic Behavior for Singularities of the Mean Curvature Flow,” J. Diff. Geom. 31 (1990) 285-299.

[I1] Ilmanen, T., “Generalized Motion of Sets by Mean Curvature on a Manifold,” Indiana J. Math. 41 (1992) 671-705.

[I2] Ilmanen, T., “The Level-Set Flow on a Manifold,” Proc. of the 1990 Summer Inst. in Diff. Geom., R. E. Greene and S. T. Yau, ed., Amer. Math. Soc., 1992.

[I3] Ilmanen, T., “Convergence of the Allen-Cahn Equation to Brakke’s Motion by Mean Curvature,” submitted.

[I4] Ilmanen, T. “Convergence to Brakke’s Motion by Mean Curvature,” Proc. of the Computational Crystal Growers Conf., J. Taylor, ed., Amer. Math. Soc., 1992.

[I5] Ilmanen, T., in preparation.

[LSU] Ladyshenskaya ?, Solonnikov ?, and Uraltseva ?, Second Order Partial Differential Equations of Parabolic Type,...

[Lu] Luckhaus, S., “Solutions for the Two-Phase Stefan Problem with the Gibbs-Thompson Law for the Melting Temperature,” European J. Appl. Math. 1 (1990) 101-111.

[M] Morgan, F., Geometric Measure Theory: A Beginner’s Guide, Academic Press, 1988.

[MS1] de Mottoni, P., and Schatzman, M., “Evolution Geometrique d’Interfaces,” C. R. Acad. Sci. Paris, Ser I 309 (1989) 453-458.

[MS2] de Mottoni, P., and Schatzman, M., “Development of Interfaces in R^{n},” Proc. Royal Soc. Edin. 116A (1990) 207-220.

[MS3] de Mottoni, P., and Schatzman, M., “Geometrical Evolution of Developed Interfaces,” Trans. Amer. Math. Soc., to appear.

[OS] Osher, S., and Sethian, J. A., “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations,” J. Computational Phys. 79 (1988) 12-49.

[S] Simon, L., Lectures on Geometric Measure Theory, Proc. Centre Math. Analysis, Austr. Nat. Univ., Vol. 3, 1983.

[SW] Solomon, B., and White, B., “A Strong Maximum Principle fo Varifolds that are Stationary with Respect to Even Parametric Elliptic Functionals,” Indiana J. Math. 38 (1989) 683-691.

[So] Soner, H. M., “Motion of a Set by the Mean Curvature of its Boundary,” J. Diff. Eq., to appear.

[SS] Soner, M., and Souganidis, P. E., “Uniqueness and Singularities of Cylindrically Symmetric Surfaces Moving by Mean Curvature,” Comm. Partial Diff. Eq., to appear.

[St] Struwe, M., “On the Evolution of Harmonic Maps in Higher Dimensions,” J. Diff. Geom. 28 (1988) 485-502.

[T] Toro, T., “Functions in W^{2,2}(R^{2}) have Lipschitz Graphs,” Ph.D. thesis, Stanford Univ., 1992.