A theorem of Hopf and the Cauchy-Riemann inequality

  • Hilario Alencar
  • Manfredo do Carmo
  • Renato Tribuzy
Chapter

Abstract

In 1951, Hopf [9] published a theorem in a seminal paper on surfaces of constant mean curvature which can be stated as follows. Let a genus zero compact surface M be immersed in \({\mathbb{R}^3}\) with constant mean curvature H. Then M is isometric to the standard sphere. Hopf gave two proofs of this result (see [9] for details).

Keywords

Quadratic Form Neighborhood Versus Ambient Space Curvature Vector Compact Surface 
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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hilario Alencar
    • 1
    • 2
    • 3
  • Manfredo do Carmo
    • 1
    • 2
    • 3
  • Renato Tribuzy
    • 1
    • 2
    • 3
  1. 1.Departamento de MatemáticaUniversidade Federal de AlagoasMaceióBrazil
  2. 2.Instituto de Matemática Pura e Aplicada (IMPA)Rio de JaneiroBrazil
  3. 3.Departamento De MatemáticaUniversidade Federal Do AmazonasManausBrazil

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