Regular globally hyperbolic maximal anti‐de Sitter structures

Abstract

Let normalΣ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti‐de Sitter structures on Σ×R and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of normalΣ and as the bundle over the Teichmüller space of normalΣ whose fibre consists of meromorphic quadratic differentials with poles of order at most 2 at the punctures.

Holomorphic quadratic differentials in Teichmüller theory

“…An alternative approach for the proof of the existence, using the method of suband super-solutions, is implicit in the work of [100] (see also §2 of [113]). In the more general context of the non-abelian Hodge correspondence mentioned in §5, the case when the Higgs field has irregular or higher order poles was considered by Biquard and Boalch in [13], where they also proved an existence theorem for the corresponding harmonic maps.…”

Degeneration of globally hyperbolic maximal anti-de Sitter structures along pinching sequences

“…Starting then from the equivariant maximal embedding into AdS 3 , it is possible to construct a maximal domain of discontinuity for the holonomy representation, thus obtaining the desired globally hyperbolic anti-de Sitter manifold as a quotient. The manifolds obtained from Theorem 1.2 are called regular (in order to distinguish them from the wild analogues in [Tam19]) and their deformation space GH reg (S p ) is thus parameterised by the bundle over Teichmüller space of S p of meromorphic quadratic differentials with poles of order at most 2 at the punctures.…”

“…. , p k }, first by allowing second order pole singularities at the punctures ( [Tam18]) and then higher order poles ( [Tam19]). We remark that, unlike the closed case, the holonomy representation does not determine completely the structure, and new extra data describing the boundary curve at infinity of the maximal surface are necessary.…”

Wild globally hyperbolic maximal anti‐de Sitter structures

“…This construction has been later generalised by the author to include non‐compact surfaces [21, 23]. In particular, if is a connected, oriented surface with punctures and negative Euler characteristic, we introduced a special class of globally hyperbolic maximal anti‐de Sitter structures on that we called regular and are parameterised by the bundle over Teichmüller space of of meromorphic quadratic differentials with poles of order at most 2 at the punctures.…”