Constant mean curvature hypersurfaces with single valued projections on planar domains

TL;DR

This paper establishes existence and uniqueness of constant mean curvature hypersurfaces with single valued projections on planar domains under new geometric conditions involving cones and mean curvature bounds.

Contribution

It extends classical results by providing new sufficient conditions involving cone containment and mean curvature bounds for hypersurfaces with prescribed mean curvature.

Findings

Proves existence and uniqueness under new geometric conditions.

Generalizes Serrin's classical result to cone-shaped domains.

Identifies conditions where the hypersurface boundary is a graph with constant mean curvature.

Abstract

A classical problem in constant mean curvature hypersurface theory is, for given $H\geq 0$, to determine whether a compact submanifold $\Gamma^{n-1}$ of codimension two in Euclidean space $\R_+^{n+1}$, having a single valued orthogonal projection on $\R^n$, is the boundary of a graph with constant mean curvature $H$ over a domain in $\R^n$. A well known result of Serrin gives a sufficient condition, namely, $\Gamma$ is contained in a right cylinder $C$ orthogonal to $\R^n$ with inner mean curvature $H_C\geq H$. In this paper, we prove existence and uniqueness if the orthogonal projection $L^{n-1}$ of $\Gamma$ on $\R^n$ has mean curvature $H_L\geq-H$ and $\Gamma$ is contained in a cone $K$ with basis in $\R^n$ enclosing a domain in $\R^n$ containing $L$ such that the mean curvature of $K$ satisfies $H_K\geq H$. Our condition reduces to Serrin's when the vertex of the cone is infinite.