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A theorem of Hopf and the Cauchy-Riemann inequality

In: Manfredo P. do Carmo – Selected Papers

Author

Listed:
  • Hilario Alencar

    (Universidade Federal de Alagoas, Departamento de Matemática
    Instituto de Matemática Pura e Aplicada (IMPA)
    Universidade Federal Do Amazonas, Departamento De Matemática)

  • Manfredo do Carmo

    (Universidade Federal de Alagoas, Departamento de Matemática
    Instituto de Matemática Pura e Aplicada (IMPA)
    Universidade Federal Do Amazonas, Departamento De Matemática)

  • Renato Tribuzy

    (Universidade Federal de Alagoas, Departamento de Matemática
    Instituto de Matemática Pura e Aplicada (IMPA)
    Universidade Federal Do Amazonas, Departamento De Matemática)

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).

Suggested Citation

  • Hilario Alencar & Manfredo do Carmo & Renato Tribuzy, 2012. "A theorem of Hopf and the Cauchy-Riemann inequality," Springer Books, in: Keti Tenenblat (ed.), Manfredo P. do Carmo – Selected Papers, edition 127, pages 441-456, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-25588-5_32
    DOI: 10.1007/978-3-642-25588-5_32
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