Summary: A HOPF DIFFERENTIAL
FOR CONSTANT MEAN CURVATURE SURFACES
IN S2 × R AND H2 × R
UWE ABRESCH AND HAROLD ROSENBERG
Dedicated to Hermann Karcher on the Occasion of his 65th
Birthday
Abstract. A basic tool in the theory of constant mean curvature (cmc)
surfaces 2
in space forms is the holomorphic quadratic differential dis
covered by H. Hopf. In this paper we generalize this differential to
immersed cmc surfaces 2
in the product spaces S2
× R and H2
× R
and prove a corresponding result about the geometry of immersed cmc
spheres S2
in these target spaces.
Introduction
In 1955 H. Hopf [13] discovered that the complexification of the traceless
part of the second fundamental form h of an immersed surface 2 with
