Nonlocal Equations on Riemannian Manifolds

Nonlocal equations have attracted a great deal of attention in recent years due to their rich analytic structure and their connections with geometry, probability, and mathematical physics. In this talk, I will discuss some recent results on nonlocal equations posed on compact Riemannian manifolds, motivated in part by the fractional Yamabe problem and related questions in conformal geometry.

I will introduce a functional framework for fractional Sobolev spaces on manifolds and explain how these spaces relate to more classical constructions from harmonic analysis. I will then present existence results for nonlinear nonlocal equations, as well as an optimal fractional Sobolev inequality in the critical regime. The emphasis of the talk will be on the geometric intuition and the variational ideas behind these problems, rather than on technical details.

This is joint work with Nicolas Saintier (Universidad de Buenos Aires).