In the mid-twentieth century, Heinz Hopf showed that the only constant mean curvature surface of genus zero is a round sphere. On the other hand Pavel Alexandrov observed that every compact embedded constant mean curvature surface is a round sphere. The question naturally arose whether compact embedded constant mean curvature surfaces existed with higher genus. This question was resolved in the affirmative by the discovery of the the first constant mean curvature torus by Henry Wente in the mid nineteen eighties [3]. His work motivated a complete classifcation of constant mean curvature tori in terms of integrable systems, and continues to inspire research in the moduli space of higher genus constant mean curvature surfaces with and without ends.

References

1. U. Abresch, Constant mean curvature tori in terms of elliptic functions, J. Reine Angew. Math. 374 (1987), 169–192.
2. Rolf Walter, Explicit examples to the H-problem of Heinz Hopf, Geom. Dedicata 23 (1987), no. 2, 187–213.
3. Henry C. Wente, Counterexample to a conjecture of H. Hopf, Pacific J. Math. 121 (1986), no. 1, 193–243.