Nirenberg's Conjecture

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 $\geq 7$ points, then the surface is a Plane.

See also Gauss Map, Minimal Surface, Neighborhood

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.