Paper detail

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

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.