Constant mean curvature hypersurfaces with single valued projections on planar domains

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.