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

Abstract

AbstractFor any $$\Lambda >0$$ Λ > 0 , let $$\mathcal {M}_{n,\Lambda }$$ M n , Λ denote the space containing all locally Lipschitz minimal graphs of dimension n and of arbitrary codimension m in Euclidean space $$\mathbb {R}^{n+m}$$ R n + m with uniformly bounded 2-dilation $$\Lambda $$ Λ 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 ∈ M n , Λ 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 $$\Lambda >1$$ Λ > 1 (we can take any $$\Lambda <\sqrt{2}$$ Λ < 2 ), we prove that (i) for $$n\le 7$$ n ≤ 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 $$\mathbb {R}^{n+m}$$ R n + m whose singular set has dimension $$\le n-7$$ ≤ n - 7 .