  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.
