Energy convexity and uniformity of the $H$-surface flow in $\mathbb{R}^{3}$ with Dirichlet boundary condition

Abstract: We prove that the energy functional associated with surfaces of prescribed mean curvature in $\mathbb{R}<sup>3$ exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form $H \cdot \text{vol}_{\mathbb{R}<sup>3}$ has a primitive satisfying certain bounds. Under milder assumptions on $H:\mathbb{R}³ \to \mathbb{R}$, we show that an analogous convexity estimate holds along the heat flow of the functional (the $H$-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the $H$-surface flow with small-energy initial map of class $C⁰ \cap W^{1, 2}$ on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the $H$-surface system) with the same boundary value.