Mean curvature flow of Reifenberg sets

Abstract

In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in ${ℝ}^{n+1}$ starting from any $n$–dimensional $\left(\epsilon ,R\right)$–Reifenberg flat set with $\epsilon$ 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 for every $n$ this class is wide enough to include some fractal sets, we obtain unique smoothing by mean curvature flow of sets with Hausdorff dimension larger than $n$, which are additionally not graphical at any scale. Except in dimension one, no such examples were previously known.