Mean Curvature Flow Of Reifenberg Sets

TL;DR

This paper proves short-term existence and uniqueness of smooth mean curvature flow starting from Reifenberg flat sets, including some fractals, showing these sets can be smoothly evolved without fattening.

Contribution

It establishes the first example of unique smoothing by mean curvature flow for sets with Hausdorff dimension greater than their topological dimension.

Findings

Flow is non-fattening and smooth for Reifenberg flat sets.

Includes fractal sets within the class of initial conditions.

Provides the first example of smoothing for high-dimensional fractal sets.

Abstract

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $\mathbb{R}^{n+1}$ starting from any $n$-dimensional $(\varepsilon,R)$-Reifenberg flat set with $\varepsilon$ sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and smooth. These sets have a weak metric notion of tangent planes at every small scale, but the tangents are allowed to tilt as the scales vary. As this class is wide enough to include some fractal sets, this provides the first example (when $n>1$) of unique smoothing by mean curvature flow of sets with Hausdorff dimension $> n$.