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

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$\acute{\mathrm{e}}$ 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 $n\le 7$, $M$ 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 $\le n-7$.

56 pages