The anisotropic Bernstein problem

TL;DR

This paper solves the anisotropic Bernstein problem by constructing nonlinear entire minimal graphs in four dimensions, extending to higher dimensions and clarifying known examples with diverse growth rates.

Contribution

It provides the first explicit constructions of nonlinear entire anisotropic minimal graphs in four dimensions, generalizing to higher dimensions and elucidating existing examples.

Findings

Constructed nonlinear entire anisotropic minimal graphs in $\,\mathbb{R}^4$.

Extended the approach to higher dimensions.

Clarified and unified known examples of entire minimal graphs.

Abstract

We construct nonlinear entire anisotropic minimal graphs over $\mathbb{R}^4$, completing the solution to the anisotropic Bernstein problem. The examples we construct have a variety of growth rates, and our approach both generalizes to higher dimensions and recovers and elucidates known examples of entire minimal graphs over $\mathbb{R}^{n},\, n \geq 8$.