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

Abstract

For any Λ > 0, let M n,Λ denote the space containing all locally Lipschitz minimal graphs of dimension n and of arbitrary codimension m in Euclidean space R n+m with uniformly bounded 2-dilation 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 ∈ 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 Λ > 1(we can take any Λ < √ 2), we prove that (i) for n ≤ 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 R n+m whose singular set has dimension ≤ n − 7.