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

Kenneth A. Brakke
Pages: 263
https://www.jstor.org/stable/j.ctt130hk4d

1. Front Matter
(pp. [i]-[ii])
(pp. [iii]-[iv])
3. 1. Introduction
(pp. 1-5)

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...

4. 2. Preliminaries
(pp. 6-17)

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

We denote by$\underline{\underline{\text{N}}}$the positive integers and by$\underline{\underline{\text{R}}}$the real numbers. Throughout this paper k and n are fixed positive integers with k ≤ n. Define${{\underline{\underline{\text{R}}}}^{+}}=\{\text{t}\in \underline{\underline{\text{R}}}:\text{t}\ge 0\},$${{\underline{\underline{\text{U}}}}^{\text{k}}}(\text{a},\text{r})=\{\text{x}\in {{\underline{\underline{\text{R}}}}^{\text{k}}}:|\text{x}-\text{a}|<\text{r}\},$$\underline{\underline{\text{U}}}(\text{a},\text{r})=\{\text{x}\in {{\underline{\underline{\text{R}}}}^{\text{n}}}:|\text{x}-\text{a}|<\text{r}\},$${{\underline{\underline{\text{B}}}}^{\text{k}}}(\text{a},\text{r})=\{\text{x}\in {{\underline{\underline{\text{R}}}}^{\text{k}}}:|\text{x}-\text{a}|<\text{r}\},$$\underline{\underline{\text{B}}}(\text{a},\text{r})=\{\text{x}\in {{\underline{\underline{\text{R}}}}^{\text{n}}}:|\text{x}-\text{a}|<\text{r}\}.$

Frequently, we will treat${{\underline{\underline{\text{R}}}}^{\text{k}}}$as a subspace of${{\underline{\underline{\text{R}}}}^{\text{n}}}$.

We will use ∫ dx to denote integration with respect to Lebesgue measure ℒnon${{\underline{\underline{\text{R}}}}^{\text{n}}}$. Set$\underline{\underline{\alpha}}={\cal {L}^{\text{k}}}{{\underline{\underline{\text{B}}}}^{\text{k}}}(0,1).$

We denote by ℋkHausdorff k-dimensional measure on${{\underline{\underline{\text{R}}}}^{\text{n}}}$.

We will often use$\left\langle \text{f},\text{g} \right\rangle$to denote the value of a...

5. 3. Motion by mean curvature
(pp. 18-32)

On first considering the problem of a surface moving by its mean curvature, one is likely to try to apply results from the theory of partial differential equations. In what is called the parametric approach, the moving surface is viewed as a family of maps${{\text{F}}_{\text{t}}}:{{\underline{\underline{\text{R}}}}^{\text{k}}}\to {{\underline{\underline{\text{R}}}}^{\text{n}}}$. From differential geometry [SM, p. 193], the mean curvature vector ht(x) at Ft(x) is the invariant Laplacian of the position vector:

(1) ht(x) = ∆Ft(x).

In coordinates, this is$\caption {(2)} {{\text{h}}_{\text{t}}}{{(\text{x})}_{\text{m}}}=\sum\limits_{\text{i},\text{j}=1}^{\text{k}}{\frac{1}{\text{g}(\text{x})}}\frac{\partial }{\partial {{\text{x}}_{\text{i}}}}[\text{g}(\text{x}){{\text{g}}^{\text{ij}}}(\text{x})\frac{\partial {{\text{F}}_{\text{t}}}{{(\text{x})}_{\text{m}}}}{\partial {{\text{x}}_{\text{j}}}}],\,\text{m}=1,\ldots,\text{k},$where (gij) is the inverse matrix of the metric (gij),${{\text{g}}_{\text{ij}}}(\text{x})=\sum\limits_{\text{p}=1}^{\text{n}}{\frac{\partial {{\text{F}}_{\text{t}}}{{(\text{x})}_{\text{p}}}}{\partial {{\text{x}}_{\text{i}}}}}\frac{\partial {{\text{F}}_{\text{t}}}{{(\text{x})}_{\text{p}}}}{\partial {{\text{x}}_{\text{j}}}},$and g2= |det(gij)|. Thus, the problem becomes to solve$\caption {(3)}\partial {{\text{F}}_{\text{t}}}(\text{x})/\partial \text{t}=\vartriangle {{\text{F}}_{\text{t}}}(\text{x}).$

This...

6. 4. Existence of varifolds moving by their mean curvature.
(pp. 33-120)

In this chapter we construct for a certain type of initial varifold V0a one parameter family of varifolds Vtdefined for all$\text{t}\in {{\underline{\underline{\text{R}}}}^{+}}$and satisfying the necessary condition for motion by mean curvature given in 3.3:$\overline{\text{D}}\left\| {{\text{V}}_{\text{t}}} \right\|(\psi)\le \delta({{\text{V}}_{\text{t}}},\psi)(\underline{\underline{\text{h}}}({{\text{V}}_{\text{t}}},.))$for any$\psi \in \underline{\underline{\text{C}}}_{0}^{1}({{\underline{\underline{\text{R}}}}^{\text{n}}},{{\underline{\underline{\text{R}}}}^{+}})$and for all$\text{t}\in {{\underline{\underline{\text{R}}}}^{+}}$. As 4.15 shows, in case V0is a smooth manifold, the construction given here agrees with the more straightforward mapping approach described in 3.1, as long as the latter works. The key properties of the present construction are proven in the last section of this chapter.

We wish to include noncompact surfaces in our...

7. 5. Perpendicularity of mean curvature.
(pp. 121-160)

We shall show in this chapter that if V is an integral varifold and ∥δV∥ is a Radon measure, then the mean curvature vector$\underline{\underline{\text{h}}}(\text{V},\text{x})$is perpendicular to the varifold at ∥V∥ almost all x. This says nothing about singular first variation, but there will be no singular first variation present in our applications in chapter 6.

One may think of the mean curvature vector as pointing in the direction of increasing mass. On a smooth manifold, mass does not increase in any tangential direction because of the local flatness; hence the mean curvature vector is perpendicular to the manifold....

8. 6. Regularity
(pp. 161-223)

In this chapter we investigate the regularity of integral varifolds moving by their mean curvature. Because of the close relationship to parabolic partial differential equations, in particular the heat equation, one would expect that such a varifold would be an infinitely differentiable manifold, except perhaps on a set of ℋkmeasure zero where several sheets join.

We shall prove in 6.13 that an integral varifold moving by its mean curvature has the desired regularity, but only under the hypothesis that the varifold has unit density almost everywhere at almost all times. Indeed, it is not even known if a stationary...

9. Appendix A: Grain growth in metals
(pp. 224-228)
10. Appendix B: Curves in R2
(pp. 229-234)
11. Appendix C: Curves of constant shape
(pp. 235-237)
12. Appendix D: Density bounds and rectiflability
(pp. 238-239)
13. FIGURE CAPTIONS
(pp. 240-240)
14. Figures
(pp. 241-250)
15. REFERENCES
(pp. 251-252)
16. Back Matter
(pp. 253-253)