Critical points of Green's functions on complete manifolds

TL;DR

This paper investigates the critical points of Green's functions on complete manifolds, establishing topological bounds in 2D and demonstrating unboundedness in higher dimensions through explicit constructions.

Contribution

It provides a topological upper bound for critical points on surfaces and constructs examples in higher dimensions showing no such bounds exist.

Findings

Topological upper bound for critical points on surfaces of finite type.

Existence of manifolds with arbitrarily many critical points in dimensions ≥ 3.

Construction of manifolds with Green's functions having level sets diffeomorphic to any codimension 1 submanifold.

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 manifold diffeomorphic to $\RR^n$ ($n \geq 3$) whose minimal Green's function has at least $N$ nondegenerate 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 $\RR^n$.