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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 the only smooth embedded minimal hypersurfaces which are topological -dimensional spheres?
Additionally, the spherical Bernstein's problem, while itself a generalization of the original Bernstein's problem, can also be generalized further by replacing the ambient space with a simply connected, compact symmetric space. Some results in this direction are due to Wu-Yi Hsiang, Wu-Teh Hsiang, and P. Tomter.
Below are two alternative ways to express the problem:
Let the (n − 1) sphere be embedded as a minimal hypersurface in (1). Is it necessarily an equator?
By the Almgren–Calabi theorem, it's true when n = 3 (or n = 2 for the 1st formulation).
Wu-Yi Hsiang proved it for n ∈ {4, 5, 6, 7, 8, 10, 12, 14} (or n ∈ {3, 4, 5, 6, 7, 9, 11, 13}, 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 -sphere is necessarily an equator?
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.
