Spherical Bernstein's problem

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.

The problem

Are the equators in [math]\displaystyle{ \mathbb{S}^{n+1} }[/math] the only smooth embedded minimal hypersurfaces which are topological [math]\displaystyle{ n }[/math]-dimensional spheres?

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 [math]\displaystyle{ \mathbb{S}^{n+1} }[/math] by a simply-connected, compact symmetric space. Some results in this direction are due to Wu-Chung Hsiang and Wu-Yi Hsiang work.

Alternative formulations

Below are two alternative ways to express the problem:

The second formulation

Let the (n − 1) sphere be embedded as a minimal hypersurface in [math]\displaystyle{ S^n }[/math](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-Chung 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)

The third formulation

Is it true that an embedded, minimal hypersphere inside the Euclidean [math]\displaystyle{ n }[/math]-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.

Further reading

• F.J. Almgren, Jr., Some interior regularity theorems for minimal surfaces and an extension of the Bernstein's theorem, Annals of Mathematics, volume 85, number 1 (1966), pp. 277–292
• E. Calabi, Minimal immersions of surfaces in euclidean spaces, Journal of Differential Geometry, volume 1 (1967), pp. 111–125
• P. Tomter, The spherical Bernstein problem in even dimensions and related problems, Acta Mathematica, volume 158 (1987), pp. 189–212
• S.S. Chern, Brief survey of minimal submanifolds, Tagungsbericht (1969), Mathematisches Forschungsinstitut Oberwolfach
• S.S. Chern, Differential geometry, its past and its future, Actes du Congrès international des mathématiciens (Nice, 1970), volume 1, pp. 41–53, Gauthier-Villars, (1971)
• W.Y. Hsiang, W.T. Hsiang, P. Tomter, On the existence of minimal hyperspheres in compact symmetric spaces, Annales Scientifiques de l'École Normale Supérieure, volume 21 (1988), pp. 287–305