Over the last few years fractional calculus has been thoroughly investigated, proving to be an effective tool for describing many physical phenomena where long range (non-local) interactions are present.

In this talk we are going to discuss the notion of *fractional s-perimeter* on a (possibly weighted) Riemannian manifold: we will show that the behaviour of such a quantity in the limit as $s\to 0^+$ depends on whether the measure of the manifold is finite or not. Such an analysis has been already performed by Figalli et al. in the Euclidean space ('12), while for the Gaussian space it has been carried out by Pallara et al. ('22).

All of our analysis is performed via a careful study of the *heat kernel*, which turns out to be the main tool to derive interesting geometric and functional properties of the $s$-perimeter.

This is a joint work with Michele Caselli.