In the mathematical field of differential geometry, the Osserman–Xavier–Fujimoto theorem concerns the Gauss maps of minimal surfaces in the three-dimensional Euclidean space.
This can be rephrased to say that the Gauss map of a complete immersed minimal surface in R3 is either constant or not contained within an open hemisphere.
As conjectured by Louis Nirenberg and proved by Robert Osserman in 1959, in this form Bernstein's theorem can be generalized to say that the image of the Gauss map of a complete immersed minimal surface in R3 either consists of a single point or is dense within the sphere.
[2] This is optimal, since it was shown by Konrad Voss in the 1960s that for any subset A of the sphere whose complement consists of zero, one, two, three, or four points, there exists a complete immersed minimal surface in R3 whose Gauss map has image A.
It is also possible to study the Gauss map of minimal surfaces of higher codimension in higher-dimensional Euclidean spaces.