Energy Convexity for Harmonic and Biharmonic Maps and H-Surfaces

In this talk, we first review energy convexity results for weakly harmonic and biharmonic maps, and then present a recent joint work with Da Rong Cheng and Xin Zhou on a convexity property of the energy functional for surfaces of prescribed mean curvature (H-surfaces) in R^3 with prescribed Dirichlet boundary data, yielding quantitative uniqueness. We also discuss energy convexity along the heat flow for H-surfaces in R^3 under a small initial Dirichlet energy assumption, leading to new results on the existence of weak solutions, long-time existence, and uniform convergence to a solution of the H-surface system with prescribed boundary conditions.