Annals of Global Analysis and Geometry

, Volume 13, Issue 1, pp 23–30 | Cite as

Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary

  • Barbara Nelli
  • Harold Rosenberg
Article

Abstract

We consider embedded hypersurfacesM in hyperbolic space with compact boundaryC and somerth mean curvature functionHr a positive constant. We investigate when symmetries ofC are symmetries ofM. We prove that if 0≤Hr≤1 andC is a sphere thenM is a part of an equidistant sphere. Forr=1 (H1 is the mean curvature) we obtain results whenC is convex.

Key words

Symmetries of hypersurfaces of constant curvature Alexandrov reflection flux formula 

MSC 1991

53A10 53C42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B]
    Barbosa, J.L.: Constant Mean Curvature Surfaces Bounded by a Planar Curve.Matematica Contemporanea 1 (1991), 3–15.Google Scholar
  2. [BJ]
    Barbosa, J.L.; Jorge, L.P.: StableH-Surfaces SpanningS 1 (1). To appear in:An. Acad. Bras. Ciênc. (1994).Google Scholar
  3. [BMRS]
    Brito, F.;Meeks III, W.H.;Rosenberg, H.;Sa Earp, R.: Structure Theorems for Constant Mean Curvature Surfaces Bounded by a Planar Curve.Indiana Univ. Math. J. 40 (1991) 1, 333–343.Google Scholar
  4. [H]
    Hopf, H.:Differential Geometry in the Large. Lecture Notes in Mathematics 1000, Springer-Verlag, 1983.Google Scholar
  5. [G]
    de Miranda Gomes, J.:Sobre hipersuperficies com curvatura media constante no espaco hiperbolico. PhD thesis IMPA, 1985.Google Scholar
  6. [K]
    Kapouleas, N.: Compact Constant Mean Curvature Surfaces in Euclidean Three-Space.J. Differ. Geom. 33 (1991), 683–715.Google Scholar
  7. [NS]
    Nelli, B.; Sa Earp, R.: Some Properties of Hypersurfaces of Prescribed Mean Curvature in ℍn+1. To appear in:Bull. Sci. Math., II. Sér.Google Scholar
  8. [R]
    Rosenberg, H.: Hypersurfaces of Constant Curvature in Space Forms.Bull. Sci. Math., II. Sér. 117 (1993), 211–239.Google Scholar
  9. [RS]
    Rosenberg, H.; Spruck, J.: On the Existence of Convex Hypersurfaces of Constant Gauss Curvature in Hyperbolic Space. To appear in:J. Differ. Geom. Google Scholar
  10. [S]
    Spivak, M.:A Comprehensive Introduction to Differential Geometry IV. Publish or Perish Inc., Berkley 1979.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Barbara Nelli
    • 1
  • Harold Rosenberg
    • 2
  1. 1.Dipartimento di MatematicaPisaItaly
  2. 2.Département de MathématiquesUniversité de Paris VIIParisFrance

Personalised recommendations