Gradient dynamical systems on open surfaces and critical points of Green's functions

TL;DR

This paper investigates the gradient flow of Green's functions on open surfaces, revealing how it decomposes the surface into a disk and a topological skeleton, and establishing bounds on critical points based on surface topology.

Contribution

It introduces a dynamical systems perspective to analyze Green's functions, providing a topological decomposition and bounds on critical points related to surface topology.

Findings

Decomposition of open surfaces into a disk and a 1-skeleton via gradient flow

Topological upper bounds on the number of Green's function critical points

Connections between gradient dynamics and surface conformal structure

Abstract

We study the dynamics of the vector field on an open surface given by the gradient of a Green's function. This dynamical approach enables us to show that this field induces an invariant decomposition of the surface as the union of a disk and a 1-skeleton that encodes the topology of the surface. We analyze the structure of this 1-skeleton, thereby obtaining, in particular, a topological upper bound for the number of critical points a Green's function can have. Connections between the dynamical properties of the gradient field and the conformal structure of the surface are also discussed.