A sharp Bernstein-type theorem for entire minimal graphs

TL;DR

This paper proves a sharp Bernstein-type theorem for entire minimal graphs in high dimensions, showing that boundedness of certain derivatives implies the solution is affine.

Contribution

It establishes a new sharp condition in high dimensions under which entire minimal graphs must be affine, extending classical Bernstein results.

Findings

If $N-7$ partial derivatives are bounded on one side, then the solution is affine.

The result applies to solutions in dimensions $N \\ge 8$.

The theorem sharpens understanding of the structure of minimal graphs in high dimensions.

Abstract

We consider entire solutions $u$ to the minimal surface equation in $R^N$, with $ N\ge8,$ and we prove the following sharp result : if $N-7$ partial derivatives $ \frac{\partial u }{\partial {x_j}}$ are bounded on one side (not necessarily the same), then $u$ is necessarily an affine function.