A theorem of Hopf and the Cauchy-Riemann inequality

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.

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Published 1 January 2007