Abstract
A general method is developed for finding necessary conditions for a given codimension-two submanifold Γ of a riemannian manifold to be the boundary of an immersed hypersurface of prescribed mean curvature. In the simplest case the condition is a comparison of the magnitude of the mean curvature with the ratio of the volume of the projection of Γ into a hyperplane to the volume of the interior of that projection. The method is applied to show that certain recent existence results for surfaces of prescribed mean curvature may not be quantitatively improved.
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References
Bernstein, S.: Sur les surfaces définies au moyen du leur courbure moyenne ou totale. Ann. Ec. Norm. Sup. (Ser. 3)27, 233–256 (1909)
Chern, S.S.: On the curvature of a piece of hypersurface in euclidean space. Abh. Math. Sem. Univ. Hamburg29, 77–91 (1965)
Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. To appear in Annals of Math.
Gulliver, R.: The Plateau problem for surfaces of prescribed mean curvature in a riemannian manifold. To appear in J. Differential Geometry
Gulliver, R., Lesley, F. D.: On boundary branch points of minimizing surfaces. To appear in Arch. Rat. Mech. Anal.
Gulliver, R., Osserman, R., Royden, H. L.: A theory of branched immersions of surfaces. Preprint
Gulliver, R., Spruck, J.: The Plateau problem for surfaces of prescribed mean curvature in a cylinder. Inventiones Math. 13, 169–178 (1971)
Gulliver, R., Spruck, J.: Existence theorems for parametric surfaces of prescribed mean curvature. Indiana U. Math. J.22, 445–472 (1972)
Heinz, E.: Über Flächen mit eineindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind. Math. Ann.129, 451–454 (1955)
Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Rat. Mech. Anal.35, 249–252 (1969)
Heinz, E.: Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern. Math. Z.113, 99–105 (1970)
Hildebrandt, S.: Randwertprobleme für Flächen mit vorgeschriebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie I. Math. Z.112, 205–213 (1969)
Hildebrandt, S., Kaul, H.: Two-dimensional variational problems with obstructions, and Plateau's problem for H-surfaces in a Riemannian manifold. Comm. Pure Appl. Math.25, 187–223 (1972)
Katsurada, Y.: On a piece of hypersurface in a Riemannian manifold with mean curvature bounded away from zero. Trans. Amer. Math. Soc.129, 447–457
Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Phil. Trans. Royal Soc. London (Ser.A)264, 413–496 (1969)
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Gulliver, R. On the nonexistence of a hypersurface of prescribed mean curvature with a given boundary. Manuscripta Math 11, 15–39 (1974). https://doi.org/10.1007/BF01189089
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DOI: https://doi.org/10.1007/BF01189089