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

TL;DR

This paper establishes a new, sharper condition for the existence of smooth solutions to the Dirichlet problem for minimal surface systems in arbitrary dimensions and codimensions, using spacelike mean curvature flow techniques.

Contribution

It introduces a novel condition for solution existence that improves upon previous results, applicable to arbitrary codimension minimal surface equations.

Findings

Derived a new existence condition for smooth solutions

Showed the condition is sharper than previous results under certain hyperbolic angle constraints

Applied spacelike mean curvature flow to analyze the problem

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 $\max_{M_{0}}{\rm cosh}\theta>\sqrt{2}$.