zbMATH — the first resource for mathematics

A theorem of Hopf and the Cauchy-Riemann inequality. (English) Zbl 1134.53031
U. Abresch and R. Rosenberg [Acta Math. 193, No. 2, 141–174 (2004; Zbl 1078.53053)] have extended Hopf’s theorem on constant mean curvature to three-dimensional spaces other than the space forms. The main result of this paper is Theorem 1.1 and it shows that, rather than assuming constant mean curvature, it suffices to assume an inequality on the differential of the mean curvature. In Proposition 4.1 the authors extend a result of R. Bryant [Complex analysis and a class of Weingarten surfaces, Preprint, unpublished] in connection with their main result.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
Full Text: DOI Euclid