Constant mean curvature hypersurfaces with single valued projections on planar domains |
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Authors: | M Dajczer |
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Institution: | a IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brazil b Instituto de Matematica, Univ. Federal do Rio Grande do Sul, Av. Bento Gonçalves 9500, 91501-970 Porto Alegre, RS, Brazil |
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Abstract: | 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. |
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