Rotation Hypersurfaces in Spaces of Constant Curvature

  • M. do Carmo
  • M. Dajczer
Chapter

Abstract

Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the helicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space H3 ; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in H3.

Keywords

Minimal Surface Fundamental Form Hyperbolic Space Constant Curvature Principal Curvature 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • M. do Carmo
    • 1
  • M. Dajczer
    • 1
  1. 1.lnstituto de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrasil

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