Constant mean curvature hypersurfaces with single valued projections on planar domains

Abstract

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