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

We prove that measurable sets E⊂ Rⁿ with locally finite perimeter and zero s-mean curvature satisfy the surface density estimates: {align*} Per (E; B_R(x)) ≥ CRⁿ⁻¹ {align*} for all $R>0$, x∈ ∂^ E. The C depends only on n and s, and remains bounded as s→ 1⁻. As an application, we prove that the fractional Sobolev inequality holds on the boundary of sets with zero s-mean curvature.