A New Way to Dirichlet Problems for Minimal Surface Systems in Arbitrary Dimensions and Codimensions

Abstract

Abstract. In this paper, by considering a special case of the spacelike mean curvature flow investigated by Li and Salavessa [Math. Z. 269 (2011), 697-719], we obtain 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 [Comm. Pure Appl. Math. 57 (2004), 267-281] provided that the hyperbolic angle θ of the initial spacelike submanifold M 0 satisfies max M 0 cosh θ > √ 2.

Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold $M^{n}\times\mathbb{R}$

“…Specially, when m = 1, the existence of ψ was obtained by Jenkins and Serrin [14] already. Inspired by Wang's work mentioned above, by applying the spacelike MCF in the Minkowski space R n+m,n , Mao [17] can successfully get the existence of ψ for maximal spacelike submanifolds (with index n) in R n+m,n . 0 In the early study of the theory of MCF, a classical result from Huisken [11] says that a given compact strictly convex hypersurface M n in R n+1 evolving along the MCF would contract to a single point at finite time.…”