Anti-de Sitter strictly GHC-regular groups which are not lattices

Compact anti-de Sitter 3-manifolds and folded hyperbolic structures on surfaces

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a surface group into PSL(2,R) is strictly dominated by some Fuchsian representation j, in the sense that the hyperbolic translation lengths for j are uniformly larger than for rho; conversely, any Fuchsian representation j strictly dominates some non-Fuchsian representation rho, whose Euler class can be prescribed. This has applications to compact anti-de Sitter 3…

具有凸边界的反德西特(2+1)时空在Fuchsian轨迹附近的刚性

Rigidity of anti-de Sitter (2+1)-spacetimes with convex boundary near the Fuchsian locus

本文证明了在Fuchsian轨迹附近，具有严格凸类空边界（光滑或多面体）的整体双曲紧致反德西特(2+1)时空由边界诱导度量唯一确定。

The Volume of complete anti-de Sitter 3-manifolds

Up to a finite cover, closed anti-de Sitter $3$-manifolds are quotients of $\mathrm{SO}_0(2,1)$ by a discrete subgroup of $\mathrm{SO}_0(2,1) \times \mathrm{SO}_0(2,1)$ of the form \[j\times ρ(Γ)~,\] where $Γ$ is the fundamental group of a closed oriented surface, $j$ a Fuchsian representation and $ρ$ another representation which is "strictly dominated" by $j$. Here we prove that the volume of such a quotient is proportional to the sum of the Euler classes of $j$ and $ρ$. As a consequence, we obtain that this volume is constant under deformation of the anti-de Sitter structure. Our results ext…

Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isomet…

几乎严格支配与反德西特3-流形

Almost strict domination and anti-de Sitter 3-manifolds

本文定义了表示对的“几乎严格支配”条件，通过求解无限能量调和映射的变分问题证明其等价于伪黎曼流形中的类空极大曲面存在性，并给出了反德西特3-流形形变空间的参数化。