A spherical Bernstein theorem for minimal submanifolds of higher codimension

TL;DR

This paper establishes spherical and Euclidean Bernstein theorems for minimal submanifolds of any dimension and codimension, using geometric analysis, Jordan angles, and properties of the Gauss map.

Contribution

It introduces new Bernstein theorems for minimal submanifolds based on the Gauss image containment in Grassmannian regions, combining geometric analysis with angle properties.

Findings

Bernstein theorems for higher codimension minimal submanifolds.

Conditions on the Gauss image ensure minimal submanifold rigidity.

Utilizes subharmonic functions, Codazzi equations, and geometric measure theory.

Abstract

Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed region of a Grassmannian manifold. The proof depends on the subharmoncity of an auxiliary function, the Codazzi equations and geometric measure theory.