Maximum principles at infinity on Riemannian manifolds: an overview

Maximum principles at infinity in the spirit of H. Omori and S.T. Yau are related to a number of properties of the underlying Riemannian manifold, ranging from the realm of stochastic analysis to that of geometry and PDEs. We will survey some of these interplays, with a special emphasis on results recently obtained by the authors, and we shall move a first step in some quite new directions. We will also present crucial applications of the maximum principles both to analytic and to geometric problems. Along the way, we will take the opportunity to introduce some unanswered questions that we feel are interesting for a deeper understanding of the subject.