Let and be the Principal Curvatures, then their Mean

(1) |

(2) |

(3) |

(4) |

(5) |

If
is a Regular Patch, then the mean curvature is given by

(6) |

(7) |

The Gaussian and mean curvature satisfy

(8) |

(9) |

If **p** is a point on a Regular Surface
and
and
are tangent vectors to at **p**, then the mean curvature of at **p** is related to the Shape
Operator by

(10) |

(11) |

Wente (1985, 1986, 1987) found a nonspherical finite surface with constant mean curvature, consisting of a self-intersecting three-lobed toroidal surface. A family of such surfaces exists.

**References**

Gray, A. ``The Gaussian and Mean Curvatures.'' §14.5 in
*Modern Differential Geometry of Curves and Surfaces.* Boca Raton, FL: CRC Press, pp. 279-285, 1993.

Isenberg, C. *The Science of Soap Films and Soap Bubbles.* New York: Dover, p. 108, 1992.

Peterson, I. *The Mathematical Tourist: Snapshots of Modern Mathematics.* New York: W. H. Freeman, pp. 69-70, 1988.

Wente, H. C. ``A Counterexample in 3-Space to a Conjecture of H. Hopf.'' In *Workshop Bonn 1984,
Proceedings of the 25th Mathematical Workshop Held at the Max-Planck Institut für Mathematik, Bonn,
June 15-22, 1984* (Ed. F. Hirzebruch, J. Schwermer, and S. Suter). New York: Springer-Verlag, pp. 421-429, 1985.

Wente, H. C. ``Counterexample to a Conjecture of H. Hopf.'' *Pac. J. Math.* **121**, 193-243, 1986.

Wente, H. C. ``Immersed Tori of Constant Mean Curvature in .'' In *Variational Methods
for Free Surface Interfaces, Proceedings of a Conference Held in Menlo Park, CA, Sept. 7-12, 1985*
(Ed. P. Concus and R. Finn). New York: Springer-Verlag, pp. 13-24, 1987.

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1999-05-26