Counterexample to a conjecture of H. Hopf. (English) Zbl 0586.53003

The author proves the existence of an immersed torus in \(E^ 3\) with constant mean curvature, thereby settling a 30 year old question of H. Hopf. He adapts methods of Y. H. Weston [SIAM J. Math. Anal. 9, 1030-1053 (1978; Zbl 0402.35038)] and J. L. Moseley [ibid. 14, 934- 946 (1983; Zbl 0543.35036)] for large positive solutions of the sinh- Gordon equation. He crucially uses a continuity argument, and the actual shape of the torus remains ”unseen”. Remark: Recently U. Abresch and R. Walter imposed the additional hypothesis of planar lines of curvature for the smaller principal curvature, and could then construct very explicit examples in terms of elliptic functions as well as computer pictures of them.
Reviewer: D.Ferus


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53A05 Surfaces in Euclidean and related spaces
35J25 Boundary value problems for second-order elliptic equations
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