Codimension 1 minimal graphs in Euclidean space have been studied successfully by many mathematicians, while minimal graphs of high codimension have many differences from the situation of codimension 1, and are far away from a thorough understanding.
For a locally Lipschitz map f: R^n→R^m, f is said to have 2-dilation ≤Λ for some constant Λ≥0 if vol(f(S))≤Λ vol(S) for each 2-dimensional open set S. Let M_(n,Λ) denote a space containing all locally Lipschitz minimal graphs of dimension n and of arbitrary codimension m in R^(n+m) with uniformly bounded 2-dilation Λ of their graphic functions. In this talk, we will discuss how this is a natural class to extend structural results known for codimension one. As applications, we get Liouville’s theorem and Bernstein theorem. This work is joint with J. Jost and Y.L. Xin.