H. Hopf proved that a topological sphere immersed in E3with constant mean curvature is a round sphere. A. D. Aleksandrov subsequently showed that the condition on the genus could be removed if the immersion were an imbedding, and he conjectured that even this condition is not necessary. However, in 1984 Wente gave an example of an immersed torus in E3with constant mean curvature. More recently, Kapouleas constructed closed surfaces in E3with constant mean curvature for any genus g > 2. In this paper, the local behavior of the Gaussian curvature K near its zero set Z is studied. Since Kmay be viewed as the ratio of surface elements with respect to the Gauss map ∅: M→S2, it follows that Z is the singular set of ∅. The classification of singularities of harmonic maps given by J. C. Wood is utilized, as is an analysis of the sinh-Gordon equation to study the critical points of K on and near Z. As a consequence, the integral whose integrand was studied by S.-S. Chern and the first author, is shown to have interesting properties.
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