Fractional Sobolev spaces on Riemannian manifolds

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 \({\mathcal {E}}(v)=[v]^2_{H^{s/2}(M)}+\int _M F(v) \, dV\), with \(F\ge 0\), 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 Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023), which defines nonlocal minimal surfaces on closed Riemannian manifolds and shows the existence of infinitely many of them for any metric on the manifold, ultimately proving the nonlocal version of a conjecture of Yau (Ann Math Stud 102:669–706, 1982). Indeed, the definitions and results in the present work serve as an essential technical toolbox for the results in Caselli et al. (Yau’s conjecture for nonlocal minimal surfaces, arxiv preprint, 2023).