Curvature estimates for minimal submanifolds of higher codimension and small G-rank

TL;DR

This paper provides new curvature estimates and Bernstein type results for minimal submanifolds in higher codimension, focusing on cases where the Gauss map rank is at most 2, including minimal surfaces.

Contribution

It introduces novel curvature bounds and Bernstein type theorems for minimal submanifolds with low Gauss map rank, extending previous results to higher codimension.

Findings

Curvature estimates are established for minimal submanifolds with Gauss map rank ≤ 2.

Bernstein type results are proved under the same rank condition.

The results apply to minimal surfaces in Euclidean spaces of arbitrary codimension.

Abstract

We obtain new curvature estimates and Bernstein type results for minimal $n-$submanifolds in $\ir{n+m},\, m\ge 2$ under the condition that the rank of its Gauss map is at most 2. In particular, this applies to minimal surfaces in Euclidean spaces of arbitrary codimension.