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arXiv · 1304.1652

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

Alberto Enciso·Daniel Peralta-Salas
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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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Alberto Enciso, Daniel Peralta-Salas. 2013-04-05. Gradient dynamical systems on open surfaces and critical points of Green's functions. https://arxiv.org/abs/1304.1652

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