Plateau Problems for Maximal Surfaces in Pseudo-Hyperbolic Spaces

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.Comment: 85 pages, 3 figures, in the version the statement of the compactness theorem 6.1 has been made more explicit for further use in some other articl