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.

last version before publishing. Statement of Theorem 7.1 for finite Plateau problem is now only true for deformable curves. The other main theorems are unchanged

References in corpus (3)

• Planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space
• Dirichlet problem for maximal graphs of higher codimension
• Complete CMC hypersurfaces in Minkowski (n+1)-space

Cited by in corpus (2)

• Complete solutions of Toda equations and cyclic Higgs bundles over non-compact surfaces
• Length spectrum compactification of the ${\mathrm{SO}}_0(2,3)$-Hitchin component