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

TL;DR

This paper establishes density estimates and fractional Sobolev inequalities for sets with zero fractional mean curvature, extending geometric analysis in fractional perimeters.

Contribution

It proves surface density estimates and fractional Sobolev inequalities for sets with zero s-mean curvature, with bounds independent of s as it approaches 1.

Findings

Sets with zero s-mean curvature satisfy uniform perimeter density estimates.

The fractional Sobolev inequality holds on boundaries of such sets.

The perimeter bounds are independent of s as s approaches 1.

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.