Spherical Bernstein theorems for codimension 1 and 2

TL;DR

This paper extends Bernstein-type theorems to spherical minimal submanifolds of codimension 1 and 2, providing new proof strategies and generalizations of Solomon's classical results.

Contribution

It introduces a novel proof approach that generalizes Solomon's theorem to codimension 2 minimal submanifolds in spheres.

Findings

Minimal hypersurfaces with certain Gauss map properties are totally geodesic.

Extension of Bernstein theorems to codimension 2 minimal submanifolds.

New proof techniques for spherical Bernstein theorems.

Abstract

A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$, whose Gauss map omits a neighborhood of an $S^{k-1}$ equator, is totally geodesic in $S^{k+1}$. We develop a new proof strategy which can also obtain an analogous result for codimension 2 compact minimal submanifolds of $S^{k+1}$.