The spherical Bernstein's problem is a possible generalization of the original

Bernstein's problem In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero m ...

in the field of global differential geometry, first proposed by

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geom ...

in 1969, and then later in 1970, during his plenary address at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

Nice Nice ( , ; Niçard: , classical norm, or , nonstandard, ; it, Nizza ; lij, Nissa; grc, Νίκαια; la, Nicaea) is the prefecture of the Alpes-Maritimes department in France. The Nice agglomeration extends far beyond the administrative c ...

The problem

Are the equators in

\mathbb^

the only smooth embedded minimal hypersurfaces which are topological

n

-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

\mathbb^

by a simply-connected, compact symmetric space. Some results in this direction are due to

Wu-Chung Hsiang Wu-Chung Hsiang (; born 12 June 1935 in Zhejiang) is a Chinese-American mathematician, specializing in topology. Hsiang served as chairman of the Department of Mathematics at Princeton University from 1982 to 1985 and was one of the most influent ...

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

S^n

(1). Is it necessarily an equator? By the Almgren–

Calabi Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...

theorem, it's true when ''n'' = 3 (or ''n'' = 2 for the 1st formulation).

proved it for ''n'' ∈ (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

n

-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.

The problem

Alternative formulations

The second formulation

The third formulation

Further reading