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 pseudohyperbolic 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.

Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces

“…The first Theorem also gives an improvement on Ishihara bounds [Ish88]. (v) We use these Rigidity Theorems and the compactness results of [LTW20] in section 6 to give six alternative characterisations of quasiperiodic surfaces involving curvature, horofunctions, spatial distance, uniformisation, Gromov hyperbolicity and laminations, some of these being stated in Theorem B. (vi) We now move in section 7 to prove Theorem 7.1.…”

“…In this section, we describe the geometry of the Einstein Universe ∂ ∞ H 2,n of signature (1, n) and study the notion of positivity in ∂ ∞ H 2,n . Although it is not relevant at this stage that the Einstein Universe is indeed the boundary of the pseudo-hyperbolic space H 2,n , we nevertheless stick (as in [LTW20]) with the notation ∂ ∞ H 2,n .…”

“…A maximal surface in H 2,n is a spacelike surface on which the mean curvature vector vanishes everywhere. We proved in [LTW20] that every complete maximal surface Σ has a limit curve ∂ ∞ Σ in ∂ ∞ H 2,n which is a semi-positive loop and, more importantly, that every such semi-positive loop is the boundary of a unique complete maximal surface.…”

“…By construction every maximal surface Σ maps into M(n), by the map which associates (x, Σ) to x. By a compactness theorem proved in [LTW20], G acts cocompactly on M(n) when the latter is equipped with the topology of smooth convergence on every compact. We will show that Barbot surfaces give rise to a unique point called the Barbot point in M(n)/G.…”

Length spectrum compactification of the $\mathrm{SO}_{0}(2,3)$-Hitchin component

“…A key ingredient for our construction is that all metrics in Ind(S) are negatively curved. This fact has been recently proven in ( [LTW20]), but we provide here a different proof, communicated to us by Qiongling Li, based only on the analysis of the Hitchin equations. Proposition 3.2.…”