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, Neighborhood## Explore 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 Conjecture## Cite this as:

Weisstein, Eric W. "Nirenberg's Conjecture."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/NirenbergsConjecture.html