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.
Nirenberg's Conjecture
See also
Complete Minimal Surface, Gauss Map, Minimal Surface, NeighborhoodExplore with Wolfram|Alpha
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.Referenced on Wolfram|Alpha
Nirenberg's ConjectureCite this as:
Weisstein, Eric W. "Nirenberg's Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NirenbergsConjecture.html