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 ⩾ 0 , to determine whether a compact submanifold Γ 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, Γ is contained in a right cylinder C orthogonal to R n with inner mean curvature H C ⩾ H . In this paper, we prove existence and uniqueness if the orthogonal projection L n − 1 of Γ on R n has mean curvature H L ⩾ − n n − 1 H and Γ is contained in a cone K with basis in R n enclosing a domain in R n containing L n − 1 such that the mean curvature of K satisfies H K ⩾ H . Our condition reduces to Serrin's when the vertex of the cone is infinite.