The evolution of H-surfaces with a Plateau boundary condition

TL;DR

This paper studies the heat flow for surfaces with prescribed mean curvature under Plateau boundary conditions, proving existence of global solutions and their convergence to conformal solutions over time.

Contribution

It introduces an isoperimetric condition on H that guarantees global weak solutions and demonstrates their long-term convergence to classical Plateau solutions.

Findings

Existence of global weak solutions under isoperimetric conditions

Solutions sub-converge to conformal solutions over time

Provides a link between heat flow dynamics and classical Plateau problem

Abstract

In this paper we consider the heat flow associated to the classical Plateau problem for surfaces of prescribed mean curvature. We show that an isoperimetric condition on H ensures the existence of a global weak solution. Moreover, we establish that these global solutions sub-converge, as time tends to infinity, to a conformal solution of the classical Plateau problem for surfaces of prescribed mean curvature.