Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation

TL;DR

This paper extends structural results for minimal graphs of arbitrary codimension with bounded 2-dilation, proving tangent cone properties, Liouville theorems, and flatness conditions in Euclidean space.

Contribution

It introduces a class of minimal graphs with bounded 2-dilation in arbitrary codimension and establishes their tangent cone structure, Liouville theorems, and flatness criteria.

Findings

Tangent cones at infinity have multiplicity one.

Liouville theorem for minimal graphs with bounded 2-dilation.

Flatness of minimal graphs for dimensions up to 7 when dilation is small.

Abstract

For any $\Lambda>0$, let $\mathcal{M}_{n,\Lambda}$ denote the space containing all locally Lipschitz minimal graphs of dimension $n$ and of arbitrary codimension $m$ in Euclidean space $\mathbb{R}^{n+m}$ with uniformly bounded 2-dilation $\Lambda$ of their graphic functions. In this paper, we show that this is a natural class to extend structural results known for codimension one. In particular, we prove that any tangent cone $C$ of $M\in\mathcal{M}_{n,\Lambda}$ at infinity has multiplicity one. This enables us to get a Neumann-Poincar$\mathrm{\acute{e}}$ inequality on stationary indecomposable components of $C$. A corollary is a Liouville theorem for $M$. For small $\Lambda>1$(we can take any $\Lambda<\sqrt{2}$), we prove that (i) for $n\leq7$, $M$ is flat; (2) for $n>8$ and a non-flat $M$, any tangent cone of $M$ at infinity is a multiplicity one quasi-cylindrical minimal cone in $\mathbb{R}^{n+m}$ whose singular set has dimension $\leq n-7$.