Infinitesimal rigidity of cone-manifolds and the Stoker problem for
hyperbolic and Euclidean polyhedra
R. Mazzeo, and G. Montcouquiol. (2009)cite arxiv:0908.2981
Comment: 4 figures; Revised Fall 2010 according to referee remarks.
Abstract
The deformation theory of hyperbolic and Euclidean cone-manifolds with all
cone angles less then $2\pi$ plays an important role in many problems in low
dimensional topology and in the geometrization of 3-manifolds. Furthermore,
various old conjectures dating back to Stoker about the moduli of convex
hyperbolic and Euclidean polyhedra can be reduced to the study of deformations
of cone-manifolds by doubling a polyhedron across its faces. This deformation
theory has been understood by Hodgson and Kerckhoff HK when the singular
set has no vertices, and by Weiss Weiss2 when the cone angles are less
than $\pi$. We prove here an infinitesimal rigidity result valid for cone
angles less than $2\pi$, stating that infinitesimal deformations which leave
the dihedral angles fixed are trivial in the hyperbolic case, and reduce to
some simple deformations in the Euclidean case. The method is to treat this as
a problem concerning the deformation theory of singular Einstein metrics, and
to apply analytic methods about elliptic operators on stratified spaces. This
work is an important ingredient in the local deformation theory of
cone-manifolds by the second author Montcouq2, see also the concurrent
work by Weiss Weiss4.
Description
Infinitesimal rigidity of cone-manifolds and the Stoker problem for
hyperbolic and Euclidean polyhedra
%0 Generic
%1 Mazzeo2009
%A Mazzeo, Rafe
%A Montcouquiol, Gregoire
%D 2009
%K class mapping surfaces
%T Infinitesimal rigidity of cone-manifolds and the Stoker problem for
hyperbolic and Euclidean polyhedra
%U http://arxiv.org/abs/0908.2981
%X The deformation theory of hyperbolic and Euclidean cone-manifolds with all
cone angles less then $2\pi$ plays an important role in many problems in low
dimensional topology and in the geometrization of 3-manifolds. Furthermore,
various old conjectures dating back to Stoker about the moduli of convex
hyperbolic and Euclidean polyhedra can be reduced to the study of deformations
of cone-manifolds by doubling a polyhedron across its faces. This deformation
theory has been understood by Hodgson and Kerckhoff HK when the singular
set has no vertices, and by Weiss Weiss2 when the cone angles are less
than $\pi$. We prove here an infinitesimal rigidity result valid for cone
angles less than $2\pi$, stating that infinitesimal deformations which leave
the dihedral angles fixed are trivial in the hyperbolic case, and reduce to
some simple deformations in the Euclidean case. The method is to treat this as
a problem concerning the deformation theory of singular Einstein metrics, and
to apply analytic methods about elliptic operators on stratified spaces. This
work is an important ingredient in the local deformation theory of
cone-manifolds by the second author Montcouq2, see also the concurrent
work by Weiss Weiss4.
@misc{Mazzeo2009,
abstract = { The deformation theory of hyperbolic and Euclidean cone-manifolds with all
cone angles less then $2\pi$ plays an important role in many problems in low
dimensional topology and in the geometrization of 3-manifolds. Furthermore,
various old conjectures dating back to Stoker about the moduli of convex
hyperbolic and Euclidean polyhedra can be reduced to the study of deformations
of cone-manifolds by doubling a polyhedron across its faces. This deformation
theory has been understood by Hodgson and Kerckhoff \cite{HK} when the singular
set has no vertices, and by Weiss \cite{Weiss2} when the cone angles are less
than $\pi$. We prove here an infinitesimal rigidity result valid for cone
angles less than $2\pi$, stating that infinitesimal deformations which leave
the dihedral angles fixed are trivial in the hyperbolic case, and reduce to
some simple deformations in the Euclidean case. The method is to treat this as
a problem concerning the deformation theory of singular Einstein metrics, and
to apply analytic methods about elliptic operators on stratified spaces. This
work is an important ingredient in the local deformation theory of
cone-manifolds by the second author \cite{Montcouq2}, see also the concurrent
work by Weiss \cite{Weiss4}.
},
added-at = {2010-12-12T07:50:25.000+0100},
author = {Mazzeo, Rafe and Montcouquiol, Gregoire},
biburl = {https://www.bibsonomy.org/bibtex/2108705bd127c42f95993825813a0b096/uludag},
description = {Infinitesimal rigidity of cone-manifolds and the Stoker problem for
hyperbolic and Euclidean polyhedra},
interhash = {cc152f395a61212a3a8dd2be79030d54},
intrahash = {108705bd127c42f95993825813a0b096},
keywords = {class mapping surfaces},
note = {cite arxiv:0908.2981
Comment: 4 figures; Revised Fall 2010 according to referee remarks},
timestamp = {2010-12-12T07:50:25.000+0100},
title = {Infinitesimal rigidity of cone-manifolds and the Stoker problem for
hyperbolic and Euclidean polyhedra},
url = {http://arxiv.org/abs/0908.2981},
year = 2009
}