The Motion of a Surface by Its Mean Curvature. (MN-20)

by Kenneth A. Brakke

Overview

Kenneth Brakke studies in general dimensions a dynamic system of surfaces of no inertial mass driven by the force of surface tension and opposed by a frictional force proportional to velocity. He formulates his study in terms of varifold surfaces and uses the methods of geometric measure theory to develop a mathematical description of the motion of a surface by its mean curvature. This mathematical description encompasses, among other subtleties, those of changing geometries and instantaneous mass losses.

Originally published in 1978.

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Product Details

ISBN-13:	9781400867431
Publisher:	Princeton University Press
Publication date:	03/08/2015
Series:	Mathematical Notes , #20
Sold by:	Barnes & Noble
Format:	eBook
Pages:	258
File size:	4 MB

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The Motion of a Surface by Its Mean Curvature

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By Kenneth A. Brakke

PRINCETON UNIVERSITY PRESS

Copyright © 1978 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08204-2

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CHAPTER 1

Introduction

Surfaces that minimize area subject to various constraints have long been studied. Much of the inspiration for these studies has come from physical systems involving surface tension: soap films, soap bubbles, capillarity, biological cell structure, and others. So far, mathematical investigations have been mostly confined to the equilibrium states of the systems mentioned, with some study of the evolution of non-parametric hypersurfaces [LT]. This work studies in general dimensions a dynamic system: surfaces of no inertial mass drive by surface tension and opposed by frictional force proportional to velocity. The viewpoint is that of geometric measure theory.

The mean curvature vector h(V,x) of a surface V at a point x can be characterized as the vector which, when multiplied by the surface tension, gives the net force due to surfact tension at that point. For example, if V is a k-sphere with radius R centered at the origin with unit magnitude surface tension, then h(V,x) = -kx/[absolute value of x]2. Note that the magnitude of the mean curvature is larger by a factor of k than in many other definitions.

The mathematical object we wish to study may be loosely described as a family of surfaces Vt parameterized by time such that each point at each time is moved with a velocity equal to the mean curvature vector of the surface at that point at that time. A physical system exhibiting this behavior is the motion of grain boundaries in an annealing pure metal. Grain boundaries represent excess energy, and there is effectively a surface tension. It is experimentally observed that these grain boundaries move with a velocity proportional to their mean curvature. For a fuller discussion, see Appendix A.

The question arises: what do we mean by a surface? We do not wish to restrict ourselves to manifolds, firstly because a manifold may evolve singularities, and secondly because systems like grain boundaries are most interesting when they are not manifolds. For surfaces we shall take a certain class of Radon measures known as varifolds, which are defined in section 2.6. The space of varifolds includes anything one would wish to call a surface and has nice compactness properties.

Since it is impossible to follow a measure pointwise, how are we to describe the motion of a surface? We do it by describing how a measure behaves on test functions. Section 2.10 derives an expression for the rate of change of the integral of a test function when the velocity of a surface is a smooth vectorfield. Section 3.3 generalizes this expression to define when a varifold is moving by its mean curvature, even when the mean curvature is far from a smooth vectorfield.

The surface with the simplest nontrivial motion are k-dimensional spheres. Let R(t) denote the radius of a k-sphere at time t. The magnitude of the mean curvature is k/R(t), so dR(t)/dt = -k/R(t). Thus

R(t) (R(0)2 - 2kt)1/2 .

The behavior of a k-sphere turns out to be characteristic of the behavior of any k-dimensional surface in the following manner: By 3.7, if a k-dimensional surface at time t = 0 is contained in the exterior of a ball of radius (R(0)2 - 2kt)1/2 with the same center. By 3.9, the analogous statement is true of a surface contained in the interior of a ball. An immediate consequence of 3.7 is that a surface starting in a convex set always remains in that convex set.

Section 3.4 establishes bounds on the motion of a surface moving by its mean curvature, and these bounds are used in 3.10 to show that the motion is continuous except for instantaneous losses of area.

Chapter 4 addresses the problem of the existence of a surface moving by its mean curvature with a given initial surface. We consider the k-dimensional initial surfaces to be members of a class of rectifiable varifolds with a positive lower bound on the k-dimensional densities. This class includes all surfaces of interest. For such an initial surface, Chapter 4 gives an approximation procedure that yields a one parameter family of varifolds that satisfies the definition of motion by mean curvature given in 3.3 and are rectifiable at almost all times. If the initial surface has integer densities, which all familiar surfaces do, then the constructed varifolds are also integral at almost all times. These properties are proven in 4.29.

If the initial surface were a smooth manifold, then one might get a solution for a short time by the straightforward use of the theory of partial differential equations, as briefly discussed in 3.1. My procedure will yield the same result in such a case, as noted in 4.15.

For a given initial surface, the subsequent motion may be naturally not unique, as illustrated in c.4. Therefore my procedure does not strive for uniqueness.

Certain modifications may easily be made at one stage to model different types of behavior (see Remark 2 of 4.9). None of these modifications affect the results of Chapter 4.

Chapter 5 proves that the mean curvature vector is almost everywhere perpendicular to an integral varifold whenever the notion of mean curvature vector is valid. This perpendicularity is easily proved for class 2 manifolds in differential geometry, but under our very broad hypotheses, we will have to delve deep into the microstructure of varifolds. This result is not directly concerned with moving varifolds, but it is essential for chapter 6.

Chapter 6 shows that a unit density integral varifold moving by its mean curvature is an infinitely differentiable manifold almost everywhere, except perhaps when there is an instantaneous loss of area. "Unit density" means that the density of the varifold is the same everywhere. Without this hypothesis, not even minimal varifolds (those with zero mean curvature) are known to be regular. Section 6.2 describes an example illustrating the problems that arise with multiple densities. Unfortunately, the existence construction of chapter 4 has not yet been made to yield unit density varifolds. However, the regularity proof is totally independent of the source of a varifold moving by its mean curvature, and would apply, for example, to the non-parametric hypersurfaces of [LT].

Appendix A discusses metal grain boundaries, as mentioned above. Appendix B discusses smooth simple closed curves in the plane moving by their mean curvature. Appendix C discusses 1-dimensional surfaces in a plane that retain the same shape but change in size as they move by mean curvature. There are computer plots of several such surfaces. Appendix D shows the necessity of the hypothesis of positive lower bounds on densities in Chapter 4 by describing a rectifiable varifold with densities approaching zero that should turn unrectifiable as it moves by its mean curvature.

As for generalizations of this work, everything would still be valid on smooth Riemannian manifolds, when properly interpreted. Extension ot integrands much different from the area integrand would not be as easy, because Allard [AW2] has shown that essentially only the area integrand satisfies monotonicity theorems, such as 4.17, which are vital to the methods herein.

I thank my advisor, Professor Frederick J. Almgren, Jr., for his guidance, for many inspiring discussions, and especially for his never-ending optimism and encouragement . I am grateful to the National Science Foundation for support.

CHAPTER 2

Preliminaries

2.1 General definitions.

We follow the standard terminology of [FH]. Most of the definitions regarding varifolds come from [AWl] .

We denote by N the positive integers and by R the real numbers. Throughout this paper k and n are fixed positive integers with k ≤ n. Define

R+ = {t [member of] R: t ≥ O},

Uk (a,r) = {x [member of] Rk: |x - a| < r},

U(a,r) = {x [member of] Rn: |x - a| < r},

Bk (a,r) = {x [member of] Rk: |x - a| < r },

B(a,r) = {x [member of] Rn: |x - a| < r}.

Frequently, we will treat Rk as a subspace of Rn·

We will use ∫ dx to denote integration with respect to Lebesgue measure Ln on Rn. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We denote by Hk Hausdorff k-dimensional measure on Rn· We will often use to denote the value of a distribution f on an appropriate test function g, especially when dealing with convolutions.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It can be shown that |Dψ|2/ψ is bounded.

If F:R [vector] B then we define for each t [member of] R the upper derivate of F at t by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We shall also use the upper left and upper right derivates, denoted D- and D+ respectively.

2.2 The Besicovitch covering theorem.

There is a positive integer B(n) with the following property: If A is a subset of Rn and C is a family of closed balls in Rn such that each point of A is the center of a member of C, then there are disjoint subfamilies Ci, i = 1,..•, B(n), of C such that

A [subset] [union]{[union] Ci: i = 1, ..., B(n)}

For the proof, see [FH 2.8.14).

2.3. Homothety and translation.

For each r [member of] R we define the homothety µ(r) = Rn [vector] Rn by

µ(r) (x) = rx.

For each a [member of] Rn we define the translation by

Τ(a) (x) = x - a.

2.4. Densities and tangent cones.

If µ is a measure on Rn then we define the k-dimensional upper density and density of μ at a [member of] Rn by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The approximate tangent cone of μ at a [member of] Rn is Tank(µ,a) = [intersection]{c: C is a cone in Rn with vertex at a and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2.5. The grassman manifold, homomorphisms, and cylinders.

Let G(n,k) denote the space of k-dimensional subspaces of Rn which can also be thought of as the set of unit simple k-vectors. Suppose S [member of] G(n,k). We will also use S to denote orthogonal projection of Rn onto S.

Let Gk(Rn) = Rn x G(n,k).

For by A, B [member of] Hom(Rn, Rn, we define a scalar product A • B

by

A • B = trace (A* ο B) •

The identity element of Hom(Rn, Rn will be denoted by I. If φ [member of] C2(Rn,R) or g [member of] C1(Rn, Rn), then we will sometimes treat D (x) as an element of Rn and D2φ(x) or Dg(x) as elements of Hom(Rn, Rn). The tensor product v * w of two vectors v,w, [member of] Rn is also in Hom(Rn, Rn). The norm * on Hom(Rn, Rn is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We will frequently use the following facts about S,T [member of] G(n,k) and A,g, φ,v,w as above:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For T [member of] G(n,k), a [member of] Rn, and 0 < r < ∞, we define the cylinder

C(T,a,r) = {x [member of] Rn: |T(x-a)| ≤ r}.

2.6 Varifolds.

We say V is a k-dimensional varifold in Rn if and only if V is a Radon measure on Gk be the weakly topologized space of k-dimensional varifolds in Rn. Whenever V [member of] Vk (Rn, we define the weight of V to be the Radon measure [parallel]v[parallel] on Rn given by

[parallel]v[parallel]A = V{(x,s) [member of] Gk (Rn): x [member of A]}

whenever A is a Borel subset of Rn· We let v(·) be the [parallel]v[parallel] measurable function with values in the Radon measures on G(n,k) such that for any ψ [member of] C0 (Gk (Rn), R)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If ΒGk(Rn) [right arrow] R [is a locally V summable function, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The same notation will be used with the obvious meaning even if Β is only defined on Rn. Similarly, if A is a [parallel]v[parallel] measurable subset of Rn, we will use V [??] A to abbreviate the restriction V [??] [A x G (n,k)], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to abbreviate β (x, S) dV (x, S).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By the well-known compactness properties of Radon measures, the set of varifolds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is compact if Bi< ?? for all i and lim Ri = ??

2.7 Rectifiable and integral varifolds.

When ever E is an [??]k measurable subset of Rn which meets every compact subset Of Rn in, an ([??]k, k) rectifiable subset [FH 3.2.14], there is a naturally Associated varifold v (E) [member of Vk Rn defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever A [subset] Gk (Rn).

We say a varifold V [member of] Vk (Rn) is a k-dimensional rectifiable varifold if there are positive real numbers c1, c2, ... and [??]k measurable subsets E1, E2, ... of (Rn) which meet every compact subsets of (Rn) in an ([??]k, k) rectifiable subset such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the ci may be taken to be positive integers, then we say V is a k-dimensional integral varifold. We let

RVk (Rn) and IVk (Rn)

be the spaces of k-dimensional rectifiable and integral varifolds respectively.

2.8. Mapping of varifolds. [AF I.1(13)]

Suppose f: Rn [right arrow] Rn is a proper mapping of class 1 and V [member of] Vk (Rn). Then the varifold f# V [member of] Vk(Rn) induced by f is characterized by the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever A is a Borel subset of Gk (Rn).

Suppose f: Rn [right arrow] Rn is a proper Lipschitz map and V [member of] RVk(Rn). Then the induced varifold f# V [member of] RVk

(Rn) is characterized by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever A is a Borel subset of GVk(Rn); here the approximate differential is

apDf(x) = ([parallel]v[parallel], k) apDf (x): Tank ([parallel]v[parallel], k) [right arrow] Rn,

see [FH 3.2.16, 3.2.19, 3.2.20]. The function f#: RVk(Rn) [right arrow] RVk(Rn) is not in general continuous. One observes f#: (IVk (Rn)) ?? IVk (Rn).

2.9. First variation.

Suppose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is smooth,

for and the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

{x: ht (x) ≠ x for some t [member of] (??, ??)]

has compact closure in an open subset G of Rn. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then V [member of] Vk (Rn) such that ???? < ?? we have by [AW1 4.1]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This motivates for any V [member of] Vk (Rn) the definition of a linear function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

called the first variation of V, by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If δ V = 0, then V is called stationary. We define the total variation [parallel]δv[parallel] to be the largest Borel regular measure on Rn determined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever G is an open subset of Rn.

(Continues...)

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Table of Contents

• Frontmatter, pg. i
• Table of Contents, pg. iii
• 1. Introduction, pg. 1
• 2. Preliminaries, pg. 6
• 3. Motion by mean curvature, pg. 18
• 4. Existence of varifolds moving by their mean curvature, pg. 33
• 5. Perpendicularity of mean curvature, pg. 121
• 6. Regularity, pg. 161
• Appendix A: Grain growth in metals, pg. 224
• Appendix B: Curves in R2, pg. 229
• Appendix C: Curves of constant shape, pg. 235
• Appendix D: Density bounds and rectiflability, pg. 238
• Figure captions, pg. 240
• Figures, pg. 241
• References, pg. 251