Asymptotics as s → 0+ of the fractional perimeter on Riemannian manifolds

In this work, we study the asymptotics of the fractional Laplacian as s → 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, generalizing both the existing results [10] for Rn and [7] for the Gaussian space. In doing so, from many sets E ⊂ M , we are able to produce a bounded harmonic function associated with E, which, in general, can be non-constant.