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  "body": "\n## Counterexample\n\nThere is an immersed torus in $\\R^3$ whose mean curvature is a nonzero constant.\nSuch a torus, constructed by Wente in 1986, is called a Wente torus.\nIt is a compact [[constant-mean-curvature surface|RPBKZ7E9VE]] that is not a round sphere.\n\n## Remarks\n\nHopf asked whether every compact immersed surface of constant mean curvature in $\\R^3$ must be a round sphere.\nHopf proved this for immersed spheres, and Alexandrov proved it for embedded surfaces of any genus.\nThe Wente torus shows that without embeddedness the answer is no in genus one.\nThe torus has self-intersections, as it must by Alexandrov's theorem.\nWente'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.\n\n## Explanations\n\n### L1 human\n\nA soap bubble that does not touch itself is always a round sphere.\nFor a long time people suspected that the same holds even if the surface may pass through itself.\nWente found a doughnut-shaped surface that passes through itself and still has the same mean curvature everywhere, which settled the question.\n",
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