Mean curvature flow of Reifenberg sets

Hershkovits, Or¹

¹ Department of Mathematics, Stanford University, 380 Serra Mall, Stanford, CA 94305, United States

Geometry & topology, Tome 21 (2017) no. 1, pp. 441-484.

Résumé

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.

Détail

DOI : 10.2140/gt.2017.21.441

Classification : 53C44
Keywords: mean curvature flow, Reifenberg sets, Reifenberg flat, non-fattening

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