Fractional Sobolev spaces on Riemannian manifolds

Abstract: This article studies the canonical Hilbert energy $H^{s/2}(M)$ on a Riemannian manifold for $s\in(0,2)$, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type $$ \mathcal E(v)=[v]^2_{H^{s/2}(M)}+\int_M F(v) \, dV \,, \,\,\, F \ge 0 \,, $$ is given, which includes, in particular, the case of nonlocal $s$-minimal surfaces. Finally, we prove some estimates for the Caffarelli-Silvestre extension problem, which are of general interest. This work is motivated by a recent article by the authors, which proves the nonlocal version of a conjecture of Yau.