The Plateau problem for polygonal boundary curves in Minkowski 3-space

TL;DR

This paper extends the solution of the Plateau problem to polygonal boundary curves in Minkowski 3-space by using maxfaces, demonstrating existence results for maximal surfaces with polygonal boundaries.

Contribution

It introduces a new existence theorem for maxfaces bounded by polygonal curves in Minkowski space, generalizing previous results that required higher boundary regularity.

Findings

Any spacelike polygonal curve in generic position bounds a maxface of disk-type.

The method adapts Garnier's approach to Minkowski space, accounting for singularities.

First existence result for polygonal boundary curves in Minkowski space.

Abstract

We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada. We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one maxface of disk-type. This is a new result, since the only known result for the Plateau problem in Minkowski space (due to N. Quien) deals with boundary curves of regularity $C^{3,\alpha}$.