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Generalized Hopf differentials Uwe Abresch

Summary: Generalized Hopf differentials
Uwe Abresch
Ruhr-Universit¨at Bochum, Alemanha
e-mail: abresch@math.rub.de
July 26, 2004
joint work with: Harold Rosenberg, Univ. Paris 7
1 Introduction
A basic tool in the theory of constant mean curvature (cmc) surfaces in
space forms is the holomorphic quadratic differential discovered by Heinz
Hopf. However, for more general target spaces the (2, 0)­part of the second
fundamental form of a cmc surface fails to be holomorphic.
The basic new result in [2] is that for cmc surfaces in the product spaces
× R and H2
× R holomorphicity can be restored with the help of explicit,
geometrically defined correction terms.
Our generalized holomorphic quadratic differential is good enough to pro-
ceed along the lines of Hopf and prove that an immersed cmc sphere S2
such a product space must in fact be one of the embedded, rotationally-


Source: Abresch, Uwe - Fakultät für Mathematik, Ruhr-Universität Bochum


Collections: Mathematics