Asymptotics as $s \to 0^+$ of the fractional perimeter on Riemannian manifolds

Abstract

In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics is related to the existence of bounded harmonic functions on $M$. As a corollary, we can find the asymptotics of the fractional $s$-perimeter on (essentially) every complete manifold, generalising both the existing results for $\mathbb{R}^n$ and for the Gaussian space. In doing so, from many sets $E\subset M$ we are able to produce a bounded harmonic function associated to $E$, which in general can be non-constant.