Density estimates and the fractional Sobolev inequality for sets of zero $s$-mean curvature

Abstract

We prove that measurable sets $E\subset {\mathbb{R}}^n$ with locally finite perimeter and zero $s$-mean curvature satisfy the surface density estimates: \begin{align*} \operatorname{Per} (E; B_R(x)) \geq CR^{n-1} \end{align*} for all $R>0$, $x\in {\partial }^{\ast }E$. The $C$ depends only on $n$ and $s$, and remains bounded as $s\to 1^{-}$. As an application, we prove that the fractional Sobolev inequality holds on the boundary of sets with zero $s$-mean curvature.