On differentiable compactifications of the hyperbolic space

On differentiable compactifications of the hyperbolic plane and algebraic actions of SL(2;R) on surfaces

It is known that the hyperbolic plane admits a countable infinity of compactifications into a closed disk such that the isometric action of SL(2;R) acts analytically on the compactified space. We prove that among those compactifications, only the two most classical ones (namely the closures of Poincaré's disk and Klein's disk) are algebraic, that is to say obtained as a union of orbits of a projectivized linear representation of SL(2;R). More generally, we classify all algebraic actions of SL(2;R) on surfaces.

Kobayashi双曲复流形的度量紧化与Denjoy--Wolff定理

The metric compactification of a Kobayashi hyperbolic complex manifold and a Denjoy--Wolff Theorem

该文研究Kobayashi双曲复流形的度量紧化性质，证明其紧化的真实性与良好性，给出拟等距嵌入连续延拓判据，关联边界点纤维并表述Wolff型引理，证明满足边界发散条件的完备双曲区域的Denjoy--Wolff定理，还给出两类紧化间恒等映射连续延拓的充要条件。

连续可微度量与权重的洛伦兹分裂定理

A Lorentzian splitting theorem for continuously differentiable metrics and weights

本文通过椭圆技术与线适应曲线，将洛伦兹分裂定理推广至$C^1$度量与权重的全局双曲加权时空。

Coarse compactifications of proper metric spaces

This paper studies coarse compactifications and their boundary. We introduce two alternative descriptions to Roe's original definition of coarse compactification. One approach uses bounded functions on $X$ that can be extended to the boundary. They satisfy the Higson property exactly when the compactification is coarse. The other approach defines a relation on subsets of $X$ which tells when two subsets closure meet on the boundary. A set of axioms characterizes when this relation defines a coarse compactification. Such a relation is called large-scale proximity. Based on this foundational wor…

Boundaries and automorphisms of hierarchically hyperbolic spaces

Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathcal X$ is proper, we get a compactification of $\mathcal X$; we also prove that our construction generalizes the Gromov boundary of a hyperbolic space. In our first main set of applications, we introduce a notion of geometrical finiteness for hierarchically hyperbolic subgroups of…

Weyl chamber length compactification of the ${\rm PSL}(2,\mathbb R)\times{\rm PSL}(2,\mathbb R)$ maximal character variety

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $Γ$ of a closed hyperbolic surface $Σ$ in ${\rm PSL}(2,\mathbb R)^n$. We identify the boundary with the sphere $\mathbb P((\mathcal{ML})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $Σ$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of $\mathbb R^2$-mixed structures on $Σ$. We associate to such a structure a dual tree-graded space endowed with an $\mathbb R_+^2$-valued metric,…