Plateau Problems for Maximal Surfaces in Pseudo-Hyperbolic Spaces

TL;DR

This paper establishes existence and uniqueness results for maximal surfaces in pseudo-hyperbolic spaces, solving asymptotic and compact Plateau problems with boundary data from positive curves at infinity.

Contribution

It introduces the first solutions to asymptotic Plateau problems for spacelike maximal surfaces in pseudo-hyperbolic spaces, utilizing pseudo-holomorphic curve analysis.

Findings

Existence and uniqueness of solutions for asymptotic Plateau problem.

Development of compactness arguments using pseudo-holomorphic curves.

Application to boundary data given by limits of positive curves.

Abstract

We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the pseudo-hyperbolic space which are limits of positive curves. We also discuss a compact Plateau problem. The required compactness arguments rely on an analysis of the pseudo-holomorphic curves defined by the Gauss lifts of the maximal surfaces.