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 : Ω ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] that satisfies the minimal surface system, see equation (1.1) in §1. The graph of u is then a minimal submanifold of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]. We prove an a priori gradient bound under the assumption that the Jacobian of du : [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i "/] → [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /] on any two dimensional subspace of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] 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.