51 research outputs found
Cross Ratios and Identities for Higher Thurston Theory
We generalise in this article the Mc Shane-Mirzakhani identities in
hyperbolic geometry to arbitrary cross ratios. We give an expression of them in
the case of Hitchin representations of surface groups in PSL(n, R) in a
suitable choice of Fock-Goncharov coordinates.Comment: 64 pages, 5 figures The new version corrects many typos, sign errors
and imprecisions. The writing has been hopefully improved. The mathematical
content is identica
Minimal surfaces and particles in 3-manifolds
We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic,
anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these
manifolds admit ``nice'' foliations and explicit metrics, and whether the space
of these metrics has a simple description in terms of Teichm\"uller theory. In
the hyperbolic settings both questions have positive answers for a certain
subset of the quasi-Fuchsian manifolds: those containing a closed surface with
principal curvatures at most 1. We show that this subset is parameterized by an
open domain of the cotangent bundle of Teichm\"uller space. These results are
extended to ``quasi-Fuchsian'' manifolds with conical singularities along
infinite lines, known in the physics literature as ``massive, spin-less
particles''.
Things work better for globally hyperbolic anti-de Sitter manifolds: the
parameterization by the cotangent of Teichm\"uller space works for all
manifolds. There is another description of this moduli space as the product two
copies of Teichm\"uller space due to Mess. Using the maximal surface
description, we propose a new parameterization by two copies of Teichm\"uller
space, alternative to that of Mess, and extend all the results to manifolds
with conical singularities along time-like lines. Similar results are obtained
for de Sitter or Minkowski manifolds.
Finally, for all four settings, we show that the symplectic form on the
moduli space of 3-manifolds that comes from parameterization by the cotangent
bundle of Teichm\"uller space is the same as the 3-dimensional gravity one.Comment: 53 pages, no figure. v2: typos corrected and refs adde
Notes on a paper of Mess
These notes are a companion to the article "Lorentz spacetimes of constant
curvature" by Geoffrey Mess, which was first written in 1990 but never
published. Mess' paper will appear together with these notes in a forthcoming
issue of Geometriae Dedicata.Comment: 26 page
Anosov representations: Domains of discontinuity and applications
The notion of Anosov representations has been introduced by Labourie in his
study of the Hitchin component for SL(n,R). Subsequently, Anosov
representations have been studied mainly for surface groups, in particular in
the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this
article we extend the notion of Anosov representations to representations of
arbitrary word hyperbolic groups and start the systematic study of their
geometric properties. In particular, given an Anosov representation of
into G we explicitly construct open subsets of compact G-spaces, on which
acts properly discontinuously and with compact quotient.
As a consequence we show that higher Teichmueller spaces parametrize locally
homogeneous geometric structures on compact manifolds. We also obtain
applications regarding (non-standard) compact Clifford-Klein forms and
compactifications of locally symmetric spaces of infinite volume.Comment: 63 pages, accepted for publication in Inventiones Mathematica
A glimpse into Thurston's work
We present an overview of some significant results of Thurston and their
impact on mathematics. The final version of this paper will appear as Chapter 1
of the book "In the tradition of Thurston: Geometry and topology", edited by K.
Ohshika and A. Papadopoulos (Springer, 2020)
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