Critical points of Green's functions on complete manifolds

Abstract

We prove that the number of critical points of a Li–Tam Green’s function on a complete open Riemannian surface of finite type admits a topological upper bound, given by the first Betti number of the surface. In higher dimensions, we show that there are no topological upper bounds on the number of critical points by constructing, for each nonnegative integer $N$, a Riemannian manifold diffeomorphic to $\mathbb{R}^n\: (n \ge 3)$ whose minimal Green’s function has at least $N$ non-degenerate critical points. Variations on the method of proof of the latter result yield contractible $n$-manifolds whose minimal Green’s functions have level sets diffeomorphic to any fixed codimension 1 compact submanifold of $\mathbb{R}^n$.