Constant mean curvature hypersurfaces with single valued projections on planar domains

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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 +ⁿ⁺¹, having a single valued orthogonal projection on ⁿ, is the boundary of a graph with constant mean curvature H over a domain in ⁿ. A well known result of Serrin gives a sufficient condition, namely, Γ is contained in a right cylinder C orthogonal to ⁿ with inner mean curvature HC≥ H. In this paper, we prove existence and uniqueness if the orthogonal projection Lⁿ⁻¹ of Γ on ⁿ has mean curvature Hₗ≥-H and Γ is contained in a cone K with basis in ⁿ enclosing a domain in ⁿ containing L such that the mean curvature of K satisfies Hₖ≥ H. Our condition reduces to Serrin's when the vertex of the cone is infinite.