| Abstract: |
| The $s$-perimeter of a set $E$ is given by the $\dot{W}^{s,1}$ norm of its characteristic function for $s\in (0,1)$. The first variation of this functional gives the $s$-mean curvature $H_s$, the fractional, nonlocal analog of typical mean curvature. We show that if your initial surface is bounded between two hyperplanes, then after evolving for a fixed finite time under fractional mean curvature flow the surface becomes a Lipschitz graph. The proof is inherently nonlocal in nature, and in fact the theorem is false for classical mean curvature flow. |
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