The spherical Bernstein's problem is a possible generalization of the original Bernstein's problem in the field of global differential geometry, first proposed by Shiing-Shen Chern in 1969, and then later in 1970, during his plenary address at the International Congress of Mathematicians in Nice.
Are the equators in
Sn+1
n
Additionally, the spherical Bernstein's problem, while itself a generalization of the original Bernstein's problem, can, too, be generalized further by replacing the ambient space
Sn+1
Below are two alternative ways to express the problem:
Let the (n − 1) sphere be embedded as a minimal hypersurface in
Sn
By the Almgren–Calabi theorem, it's true when n = 3 (or n = 2 for the 1st formulation).
Wu-Chung Hsiang proved it for n ∈ (or n ∈, respectively)
In 1987, Per Tomter proved it for all even n (or all odd n, respectively).
Thus, it only remains unknown for all odd n ≥ 9 (or all even n ≥ 8, respectively)
Is it true that an embedded, minimal hypersphere inside the Euclidean
n
Geometrically, the problem is analogous to the following problem:
Is the local topology at an isolated singular point of a minimal hypersurface necessarily different from that of a disc?
For example, the affirmative answer for spherical Bernstein problem when n = 3 is equivalent to the fact that the local topology at an isolated singular point of any minimal hypersurface in an arbitrary Riemannian 4-manifold must be different from that of a disc.