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: the classical result for Rn by Dipierro-Figalli-Palatucci-Valdinoci (2012) and the recent one for the Gaussian space by Carbotti-Cito-La Manna-Pallara (2021). In doing so, from many sets E contained in M we are able to produce a bounded harmonic function associated to E, which in general can be non-constant.