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

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, first proposed by

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 26, 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 geome ...

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 IMU Abacus Medal (known before ...

Nice Nice ( ; ) is a city in and the prefecture of the Alpes-Maritimes department in France. The Nice agglomeration extends far beyond the administrative city limits, with a population of nearly one million Wu-Chung Hsiang Wu-Chung Hsiang (; born 12 June 1935) is a Taiwanese-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 influential Topolog ...

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 (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professorship of Mathematics, Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in dif ...

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

Wu-Chung Hsiang Wu-Chung Hsiang (; born 12 June 1935) is a Taiwanese-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 influential Topolog ...

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.

Alternative formulations

The second formulation

The third formulation

Further reading