Energy Convexity and Uniformity of H-surface Flow in Two Dimensions

In this talk, we present a convexity property of the energy functional for surfaces of prescribed mean curvature (also known as H-surfaces) in R3 with prescribed Dirichlet boundary data, yielding a quantitative uniqueness result for solutions to the H-surface equation. We will also discuss an energy convexity property along the heat flow for H-surfaces in R3, assuming only that the initial Dirichlet energy is sufficiently small, leading to a new theorem on the existence of weak solutions, long-time existence, and uniform convergence of the flow to a solution of the H-surface system with prescribed Dirichlet boundary conditions. This talk is based on a recent joint work with Da Rong Cheng and Xin Zhou.