Differential geometry Minimal and constant mean curvature surfaces Counterexample

Wente torus

Builds on Constant-mean-curvature surface

Counterexample

There is an immersed torus in $\R^3$ whose mean curvature is a nonzero constant. Such a torus, constructed by Wente in 1986, is called a Wente torus. It is a compact constant-mean-curvature surface that is not a round sphere.

Remarks

Hopf asked whether every compact immersed surface of constant mean curvature in $\R^3$ must be a round sphere. Hopf proved this for immersed spheres, and Alexandrov proved it for embedded surfaces of any genus. The Wente torus shows that without embeddedness the answer is no in genus one. The torus has self-intersections, as it must by Alexandrov's theorem. Wente's construction reduces the problem to doubly periodic solutions of a sinh-Gordon equation for the conformal factor, together with conditions that make the surface close up.

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Constant-mean-curvature surface concept
Wente torus counterexample

Tag	`E5NNY7XM44`
URI	`m2://E5NNY7XM44`
Kind	counterexample
Writing	W0 stub
MSC	53A10, 53C42
Uses	Constant-mean-curvature surface
Changed	2 November 2026

Wente torus

Counterexample

Remarks

Reading path

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