Stability and curvature estimates for minimal graphs with flat normal bundles

TL;DR

This paper proves stability and flatness results for higher codimensional minimal graphs with flat normal bundles, extending classical results and establishing Bernstein type theorems under growth conditions.

Contribution

It establishes stability of minimal graphs with flat normal bundles in any codimension and dimension, and proves flatness under finite density and growth conditions.

Findings

Minimal graphs with flat normal bundles are stable in any codimension.

Such graphs are flat if dimension ≤ 6 and density at infinity is finite.

Bernstein type theorem holds under additional growth assumptions.

Abstract

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minimal graphs of dimension no greater than six and any codimension is flat if the the normal bundle is flat and the density at infinity is finite. Such a Bernstein type theorem holds in any dimension if we assume additionally growth conditions on the volume element.