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

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