Constant mean curvature hypersurfaces with single valued projections on planar domains

A classical problem in constant mean curvature hypersurface theory is, for given H⩾0, to determine whether a compact submanifold Γn−1 of codimension two in Euclidean space , having a single valued orthogonal projection on Rn, is the boundary of a graph with constant mean curvature H over a domain in Rn. A well known result of Serrin gives a sufficient condition, namely, Γ is contained in a right cylinder C orthogonal to Rn with inner mean curvature HC⩾H. In this paper, we prove existence and uniqueness if the orthogonal projection Ln−1 of Γ on Rn has mean curvature and Γ is contained in a cone K with basis in Rn enclosing a domain in Rn containing Ln−1 such that the mean curvature of K satisfies HK⩾H. Our condition reduces to Serrin's when the vertex of the cone is infinite.