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Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

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Manfredo P. do Carmo – Selected Papers

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Abstract

The main result of this paper states that the traceless second fundamental tensor \({A}^{0}\)of an n-dimensional complete hypersurface M, with constant mean curvature H and finite total curvature, \(\int{M}^{|A^0|^n{d\nu}}{M}<\infty \), in a simply-connected space form \(\bar{M}(c)\), with non-positive curvature c, goes to zero uniformly at infinity. Several corollaries of this result are considered: any such hypersurface has finite index and, in dimension 2, if \(H^2+c>0,\) any such surface must be compact.

Mathematics Subject Classification (1991): 53C42, 53C21, 35J60

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Authors and Affiliations

  1. Institut Fourier, UMR 5582, Université Joseph Fourier- CNRS, 74, 38402, Saint Martin d’Hères Cedex, France

    P. Bérard

  2. I.M.P.A., Estrada Dona Castorina 110, Jardim Botânico, 22460-320, Rio de Janeiro, Brazil

    M. do Carmo

  3. Departamento de Matemática, Universidade Federal do Rio de Janeiro, 68530, 21944, Rio de Janeiro, Brazil

    W. Santos

Authors
  1. P. Bérard
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  2. M. do Carmo
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  3. W. Santos
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Corresponding author

Correspondence to P. Bérard .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Brasília, Brasilia, 70910-900, Brazil

    Keti Tenenblat

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Bérard, P., Carmo, M.d., Santos, W. (2012). Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature. In: Tenenblat, K. (eds) Manfredo P. do Carmo – Selected Papers. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25588-5_26

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  • DOI: https://doi.org/10.1007/978-3-642-25588-5_26

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  • Print ISBN: 978-3-642-25587-8

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