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

2013/04/05 by Alberto Enciso, Daniel Peralta‐Salas, Enciso, Alberto +1
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1304.1652

openalex publication_date 2013/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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