Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Rigidity of 3-cone-manifolds and the Stoker
problem
Grégoire Montcouquiol
Université Orsay Paris-Sud
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Outline of the talk
1
Some results on the rigidity of polyhedra, link with
cone-manifolds.
2
An infinitesimal rigidity result and the solution to the Stoker
problem.
3
A sketch of the proof : the relationship between cone angles,
indicial roots and regularity.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Is this a regular octahedron ?
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Earlier rigidity results
Theorem (Cauchy 1813 et al.)
Two convex polyhedra having the same combinatorial structure
and the same faces are congruent.
Also true for hyperbolic and spherical polyhedra (Pogorelov 73).
This theorem is clearly false for non-convex polyhedra:
However in this case local rigidity still holds.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Infinitesimal rigidity
Flexible (non-convex) polyhedra exist (Connelly 77) but are
exceptional.
They gave rise to the famous “bellows theorem” (Sabitov 96).
Generically, polyhedra are infinitesimally rigid (Gluck, Sabitov).
Definition
A polyhedron P is infinitesimally rigid (relatively to the shape of
its faces) if every first-order deformation preserving the shape of
the faces is trivial, i.e. comes from a rigid motion of P.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Polyhedra and cone-surfaces
The metric on the boundary of a polyhedron
- extends smoothly across its edges
- has cone-like singularities at the vertices.
Convex polyhedron
Rigidity of 3-cone-manifolds
Ñ flat cone metric on the sphere
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
More rigidity
Theorem (Alexandrov 50)
A convex euclidean polyhedron is uniquely determined by the flat
cone metric of its boundary.
Every flat cone metric on S 2 with cone angles smaller than 2π is
obtained that way.
Once again, it clearly does not hold for non-convex polyhedra:
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Stoker’s conjecture
In 1968, Stoker asked wether convex polyhedra were determined by
their dihedral angles.
It is not true in the euclidean case because of parallel motions of
the faces:
But the internal angles of the faces remain constant.
These motions do not exist in the spherical and hyperbolic cases.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Stoker’s conjecture
Conjecture (spherical and hyperbolic cases)
Convex polyhedra are infinitesimally rigid relatively to their
dihedral angles.
Conjecture (euclidean case)
Every first-order deformation of a convex polyhedron which
preserves the dihedral angles also preserves the internal angles of
the faces.
• First proofs by Karcher, Milka, Pogorelov, in special cases.
• The conjecture is false in the spherical case (Schlenker 2000).
This conjecture is related to similar statements about
cone-manifolds.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
An example of cone-manifold
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
An example of cone-manifold
singular locus
Rigidity of 3-cone-manifolds
stratification
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Curvature κ cones
We introduce a new function:
Indicial roots and regularity
$ sinp?κ x q
'
'
& ?κ ,
snκ px q x,
'
'
% sinh?p?κ x q ,
κ
κ¡0
κ0
κ 0
Let L be a (possibly singular) Riemannian space, with metric gL .
The -truncated curvature κ cone over L is the cone
r0, s L{p0,pqp0,p q
1
together with the singular Riemannian metric
g
dr 2
snκ pr q2 gL , r
P r0, s
It is of constant sectional curvature κ iff L is of constant sectional
curvature 1.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Inductive definition
Definition
A 1-cone-manifold is a dimension 1 Riemannian manifold.
A curvature κ n-cone-manifold, n ¡ 1, is a metric space in wich
each point p has a neighborhood isometric to a truncated
curvature κ cone over some closed, connected, orientable, spherical
pn 1q-cone-manifold, called the link of p.
A point p is regular if its link is the whole sphere, and singular
otherwise.
The set of singular points is called the singular locus. It is a
totally geodesic, stratified subset of the cone-manifold.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Inductive definition - an example
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Inductive definition - an example
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Inductive definition - an example
Length 2!
Length "
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Polyhedra and cone-manifold
The double of a polyhedron is naturally a cone-manifold,
homeomorphic to the sphere:
The polyhedron is convex
Rigidity of 3-cone-manifolds
ô the cone angles are smaller than 2π
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Infinitesimal rigidity
Theorem (Mazzeo-M.)
- Closed hyperbolic 3-cone-manifolds, whose angles are smaller
than 2π, are infinitesimally rigid relatively to cone angles.
- Every angle-preserving first-order deformation of a euclidean
3-cone-manifold, whose angles are smaller than 2π, preserves
the links of the vertices.
Corollary
The Stoker conjecture is true in the hyperbolic and euclidean cases.
• Hodgson-Kerckhoff 98: hyperbolic case when the singular
locus is a link.
• Weiss 02: hyperbolic and spherical case when the angles are
smaller than π, some results in the euclidean case.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Framework
Infinitesimal deformations
Let gt be a family of cone metrics on M 3 , with singular locus Σ,
constant curvature κ P t1, 0u, and g0 g .
The infinitesimal deformation h on M, satisfying
∇g ∇g h
|
dgt
dt t 0
is a symmetric 2-tensor
2κph ptr g hqg q δg p2δg h
dtr g hq 0
Definition
An infinitesimal deformation is a symmetric 2-tensor, solution of
the above equation.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Framework
Standard form
The metric near the singular locus is determined by a limited
number of parameters.
In particular any infinitesimal deformation of a cone-manifold can
be expressed near the singular locus as a linear combination of four
types of deformation, respectively :
• changing the lengths of the singular edges
• changing the “twist parameters” along the singular edges
• changing the spherical links of the vertices of the singular
locus
ë not modifying the cone angles
ë modifying the cone angles
All of the above are in L2 , with covariant derivative in L2 , except
for the last one.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Framework
Trivial deformations
Let ϕt be a family of diffeomorphisms of pM, Σq such that ϕ0
Id.
Then the metrics gt ϕt g are of constant curvature κ.
The corresponding infinitesimal deformation is
Bϕt g L g 2δη.
g
Bt |t 0 X
Definition
Any symmetric 2-tensor belonging to the range of the operator δg
is called a trivial deformation.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Gauge condition
We want to impose the Bianchi gauge condition, i.e. we want to
consider only deformations satisfying
βg h δg h
1
dtr g h 0.
2
Normalizing an infinitesimal deformation h0 consists of finding a
trivial deformation δ η such that the deformation h h0 δ η
satisfies the Bianchi gauge condition.
It amounts to solving the normalization equation β pδ η q βh0 ,
which can be expressed as
∇ ∇η κpn 1qη
Rigidity of 3-cone-manifolds
2βh0.
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Bochner technique (hyperbolic case)
1
Normalization: h h0 δ η satisfies βh 0 and
∇ ∇h 2h 2ptr hqg 0.
2
We take the trace: ∆tr h 2pn 1qtr h 0. An integration
by parts yields tr h 0, δh 0, and ∇ ∇h 2h 0.
3
We use the Weitzenböck’s formula
∇ ∇h pδ ∇ d ∇ d ∇ δ ∇ qh n h ptr hqg . We get
δ ∇ d ∇ h pn 2qh 0 and conclude with an integration by
parts.
Issue: Can we integrate by parts ?
ùñ need to control the behavior of the normalized deformation
near the singular locus
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Regularity issues
The operator P ∇ ∇
has good properties :
2 involved in the normalization equation
• symmetry
• positivity
• ellipticity
Ñ smoothing properties outside of the singularities
But because of the singularities, traditional regularity results do
not hold.
We will see that the behavior near the singular locus depends on
the cone angles.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Domains
A cone-like singularity can be seen as some kind of “degenerate (or
collapsed) boundary”.
An analog of the classical boundary conditions (i.e. Dirichlet or
Neumann conditions on a subset of Rn ) is needed.
ñ choice of a domain, i.e. a subspace D, C08 € D € L2, on
which:
• P is well-defined
• P has some good properties: self-adjointness, positivity,
Fredholm. . .
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
An example : the flat 2-dimensional case
Let Σ be a closed Euclidean cone-surface. We will investigate the
properties of the Laplacian ∆ acting on functions on Σ.
A good choice of domain for ∆ is
D
tf P L2 | Df ,
∆f
P L2u
This corresponds to solving in the Sobolev space H 1 .
On this domain, ∆ is self-adjoint, Fredholm, and its kernel is
exactly the set of constant functions.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
An example : the flat 2-dimensional case
Near a singular point x of cone angle α, one can use polar
coordinates and Fourier series decomposition: with γ 2π {α,
f pr , θ q ¸
P
n Z
∆f
fn pr q exppinγθq,
¸ 2 1 1
fn r fn
P
n Z
n2 γ 2
fn exppinγθq.
r2
The functions r nγ exppinγθq, n P Z, as well as ln r , thus satisfy
∆f 0 near x. The real numbers nγ are indicial roots of the
Laplace operator, 0 being a double root.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
An example : the flat 2-dimensional case
The choice of a domain for ∆ specifies which indicial roots can
occur.
With D tf P L2 | Df , ∆f P L2 u, only positive indicial roots are
allowed.
If there exists an indicial root k nγ 2πn{α P p0, 1q, then the
function f pr θq r k exppinγθq (suitably truncated) is such that
f , Df , ∆f P L2 but D 2 f R L2 !
This can only happen if the cone angle α is bigger than 2π.
Proposition
If all cone angles are smaller than 2π, then every function f such
that f , Df and ∆f are in L2 also satisfies D 2 f P L2 .
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
The 1-forms case
For 1-forms, there exist two canonical choices of domains, on
which the Laplace-de Rham operator ∆ dδ δd ∇ ∇ ric is
self-adjoint :
• DDN
η
P Ω1M | η,
dη, δdη, δη, dδη
Gaffney)
• DF
η
P Ω1M | η,
∇η, ∇ ∇η
P L2
(
P L2
(
(Cheeger,
(Friedrichs extension)
They give rise to the same set of indicial roots if all cone angles
are smaller than 2π.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Normalizing the deformations
If M is a closed hyperbolic 3-cone-manifold, the computation of
the indicial roots allows to prove the following theorem :
Theorem
If all cone angles are smaller than 2π, then the normalization
equation ∇ ∇η 2η φ, with φ P L2 pT M q, has a unique
solution η P L2 pT M q such that ∇η and ∇dη are in L2 .
With these properties on the normalized deformation, it is then
possible to apply the Bochner technique
ñ infinitesimal rigidity in the hyperbolic case.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
The euclidean case
The Bochner technique is a bit different in the euclidean case.
1
The equation for the trace is ∆ptr hq 0
This corresponds to scaling.
2
The resulting equations for h are
ñ tr h cst.
∇ ∇h 0
(1)
δ∇d ∇h 0
(2)
(1) cannot be integrated by parts, but (2) can.
$ ∇
'
&d h 0
ñ h P H 1pM, TM q
δ∇h 0
'
%pd ∇ q2 0
This implies that the vertices’ links are not deformed.
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Future
1
From infinitesimal rigidity to local rigidity and parametrization
by cone angles.
This requires an understanding of the splitting of vertices.
2
What if some cone angles are greater than 2π ?
3
Spherical case still mysterious.
4
Higher dimensional case (Einstein deformations).
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol
Polyhedra and cone-manifolds
Infinitesimal rigidity
Indicial roots and regularity
Rigidity of 3-cone-manifolds and the Stoker
problem
Grégoire Montcouquiol
Université Orsay Paris-Sud
Rigidity of 3-cone-manifolds
Grégoire Montcouquiol