Communications in Analysis and Geometry

Volume 15 (2007)

Number 2

A theorem of Hopf and the Cauchy-Riemann inequality

Pages: 283 – 298



Hilario Alencar

Manfredo do Carmo

Renato Tribuzy


Recently, Abresch and Rosenberg (“A Hopf differential for constant mean curvature surfaces” in $S/sp 2 x \Bbb R$ and $H/sp 2 x \Bbb R$ (U. Abresch, R. Rosenberg, Acta Math. 193 (2004), no. 2, 141-174) have extended Hopf’s Theorem on constant mean curvature to 3-dimensional spaces other than the space forms. Here we show that, rather than assuming constant mean curvature, it suffices to assume an inequality on the differential of the mean curvature.

Published 1 January 2007