A new way to Dirichlet problems for minimal surface systems in arbitrary dimensions and codimensions

Abstract

In this paper, by considering a special case of the spacelike mean curvature flow investigated by Li and Salavessa [6], we get a condition for the existence of smooth solutions of the Dirichlet problem for the minimal surface equation in arbitrary codimension. We also show that our condition is sharper than Wang's in [13, Theorem 1.1] provided the hyperbolic angle $\theta$ of the initial spacelike submanifold $M_{0}$ satisfies $\max_{M_{0}}{\rm cosh}\theta>\sqrt{2}$.