Constant mean curvature hypersurfaces with single valued projections on planar domains

A classical problem in constant mean curvature hypersurface theory is, for given $Hgeq 0$, 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 $H_Cgeq H$. In this paper, we prove existence and uniqueness if the orthogonal projection $L^{n-1}$ of $Gamma$ on $R^n$ has mean curvature $H_Lgeq-H$ 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 $H_Kgeq H$. Our condition reduces to Serrins when the vertex of the cone is infinite.

Submanifolds of codimension two attaining equality in an extrinsic inequality

We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the length of the mean curvature vector.