Interior gradient bounds for solutions to the minimal surface system

In this article we generalize the classical gradient estimate for the minimal surface equation to higher codimension. We consider a vector-valued function u : Omega subset of R-n --> R-m that satisfies the minimal surface system, see equation (1.1) in 1. The graph of u is then a minimal submanifold of Rn+m. We prove an a priori gradient bound under the assumption that the Jacobian of du : R-n --> R-m on any two dimensional subspace of R-n is less than or equal to one. This assumption is automatically satisfied when du is of rank one and thus the estimate covers the case when m = 1, i.e., the original minimal surface equation. This is applied to Bernstein type theorems for minimal submanifolds of higher codimension.