---
tag: E5NNY7XM44
kind: counterexample
name: Wente torus
msc: [53A10, 53C42]
uses: [RPBKZ7E9VE]
sources:
  - url: https://en.wikipedia.org/wiki/Wente_torus
---

## 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|RPBKZ7E9VE]] 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.

## Explanations

### L1 human

A soap bubble that does not touch itself is always a round sphere.
For a long time people suspected that the same holds even if the surface may pass through itself.
Wente found a doughnut-shaped surface that passes through itself and still has the same mean curvature everywhere, which settled the question.