If the Gauss Map of a complete minimal surface omits a Neighborhood of the Sphere, then the surface is a Plane. This was proven by Osserman (1959). Xavier (1981) subsequently generalized the result as follows. If the Gauss Map of a complete Minimal Surface omits points, then the surface is a Plane.

**References**

do Carmo, M. P. *Mathematical Models from the Collections of Universities and Museums* (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, p. 42, 1986.

Osserman, R. ``Proof of a Conjecture of Nirenberg.'' *Comm. Pure Appl. Math.* **12**, 229-232, 1959.

Xavier, F. ``The Gauss Map of a Complete Nonflat Minimal Surface Cannot Omit 7 Points on the Sphere.'' *Ann. Math.*
**113**, 211-214, 1981.

© 1996-9

1999-05-25