Convexity of constant mean curvature graphs in $\mathbb{R}^{n+1}$ with planar boundary

TL;DR

This paper proves that in Euclidean space, a graph with constant mean curvature and a convex boundary is strictly convex if the boundary's normalized mean curvature exceeds or equals the mean curvature of the graph.

Contribution

It establishes that the optimal boundary curvature condition ensures the strict convexity of constant mean curvature graphs with convex boundary.

Findings

Convex boundary with boundary curvature h ≥ H guarantees strict convexity.

The solvability condition h ≥ H is sufficient for convexity in convex domains.

Results extend understanding of geometric properties of constant mean curvature surfaces.

Abstract

We study the Dirichlet problem for a graph $\Sigma$ in $\mathbb{R}^{n+1}$ with normalized constant mean curvature $H>0$ and planar boundary $\Gamma=\partial \Omega$. Our main result is that the optimal solvability condition, namely that the normalized mean curvature $h$ of $\Gamma$ satisfies $h\geq H$, also suffices when $\Omega$ is strictly convex, to prove the strict convexity of $\Sigma$.