Deformations of Hyperbolic Cone-Structures: Study of the Collapsing case

0.47 MB·English

by  Alexandre Paiva Barreto

DEFORMATIONS OF HYPERBOLIC CONE-STRUCTURES: STUDY OF THE
COLLAPSING CASE
ALEXANDREPAIVABARRETO
Dedicated to professor Paulo Sabini (1972 - 2007)
2
1
Abstract. This work is devoted to the study of deformations of hyperbolic cone structures
0
under the assumption that the lengths of the singularity remain uniformly bounded over the
2
deformation. Given a sequence (Mi,pi) of pointed hyperbolic cone-manifolds with topological
n type (M,Σ), where M is a closed, orientable and irreducible 3-manifold and Σ an embedded
a linkinM. IfthesequenceMi collapsesandassumingthatthelengthsofthesingularityremain
J uniformlybounded,weprovethatM iseitheraSeifertfiberedoraSolmanifold. Weapplythis
3 resulttoaquestionstatedbyThurstonandtothestudyofconvergentsequencesofholonomies.
1

]
T
1. Introduction
G
. Fixed a closed, orientable and irreducible 3-manifold M, this text focus deformations of
h
t hyperbolic cone structures on M which are singular along a fixed embedded link Σ=Σ1(cid:116)...(cid:116)Σl
a
in M. A hyperbolic cone structure with topological type (M,Σ) is a complete intrinsic metric on
m
M (see section 2 for the definition) such that every non-singular point (i.e. every point in M −Σ)
[ has a neighborhood isometric to an open set of H3, the hyperbolic space of dimension 3, and that

1 every singular point (i.e. every point in Σ) has a neighborhood isometric to an open neighborhood
v of a singular point of H3(α), the space obtained by identifying the sides of a dihedral of angle
3 α∈(0,2π] in H3 by a rotation about the axe of the dihedral. The angles α are called cone angles
2
and they may vary from one connected component of Σ to the other. By convention, the complete
9
structure on M −Σ is considered as a hyperbolic cone structure with topological type (M,Σ) and
2
. cone angles equal to zero.
1
Unlike hyperbolic structures, which are rigid after Mostow, the hyperbolic cone structures can
0
be deformed (cf. [HK2]). The difficulty to understand these deformations lies in the possibility of
2
1 degenerating the structure. In other words, the Hausdorff-Gromov limit of the deformation (see
: section 2 for the definition) is only an Alexandrov space which may have dimension strictly smaller
v
i than 3, although its curvature remains bounded from below by −1 (cf. [Koj]). In fact, the works
X
of Kojima, Hodgson-Kerckhoff and Fuji (see [Koj], [HK] and [Fuj]) show that the degeneration of
r the hyperbolic cone structures occurs if and only if the singular link of these structures intersects
a
itself over the deformation.
When M is a hyperbolic manifold and Σ is a finite union of simple closed geodesics of M, it is
knowthatM−Σadmitsacompletehyperbolicstructure(see[Koj]). Weareinterestedinstudying
the following question that was raised by W.Thuston in 80’s:
2000 Mathematics Subject Classification. Primary55m20.
Thisworkispartofthedoctoralthesisoftheauthor,madeatUniversitPaulSabatierunderthesupervisionof
ProfessorMichelBoileauandsupportedbyCAPESandFAPESP.
1
2 ALEXANDREPAIVABARRETO
Question 1.1. Let M be a closed and orientable hyperbolic manifold and suppose the existence of
a simple closed geodesic Σ in M. Can the hyperbolic structure of M be deformed to the complete
hyperbolic structure on M −Σ through a path M of hyperbolic cone structures with topological
α
type (M,Σ) and parametrized by the cone angles α∈[0,2π]?
WhenthedeformationproposedbyThurstonexists,itisaconsequenceofThurston’shyperbolic
Dehn surgery Theorem that the length of Σ must converge to zero. In particular, we have that
the length of the singular link remains uniformly bounded over the deformation. Motivated by
this fact, we are interested in studying the deformations of hyperbolic cone structures with this
aditional hypothesis on the length of the singular link. This assumption is automatically verified
whentheholonomyrepresentationsofthehyperbolicconestructuresareconvergent. Thishappens
(cf. [CS]), for example, when Σ is a small link in M. Note that this additional hypothesis avoids
the undesirable case where the singularity becomes dense in the limiting Alexandrov space.
In this paper, M will always denote a sequence of hyperbolic cone-manifolds with topologi-
i
cal type (M,Σ). Given points p ∈ M , suppose that the sequence (M ,p ) converges (in the
i i i i
Hausdorff-Gromov sense) to a pointed Alexandrov space (Z,z ). Our goal is to understand the
0
metric properties of the Alexandrov space Z and exploit them to obtain further topological infor-
mation on the manifold M. To do this, we need the important notion of collapse for a sequence
and a continuous deformation of hyperbolic cone-manifolds.
Definition 1.2. We say that a sequence M of hyperbolic cone-manifolds with topological type
i
(M,Σ) collapses if, for every sequence of points p ∈M −Σ, the sequence rMi−Σ(p ) consisting of
i inj i
theirRiemannianinjectivityradiiinM −Σconvergestozero. Otherwise,wesaythatthesequence
i
M does not collapse.
i
Whenaconvergentsequenceofhyperboliccone-manifoldsdoesnotcollapse,thelimitAlexandrov
space must have dimension 3. In this case, several geometric techniques are known to study the
topological type of M. On the collapsing case, however, most of the geometric information can be
lost. This happens because the dimension of the limit Alexandrov space may be strictly smaller
than 3 (see 3.4). Therefore, finding conditions that eliminate the possibility of collapse is very
useful.
Denote by L (Σ ) the length of the connected component Σ of Σ in the hyperbolic cone-
Mi j j
manifold M . Using this notation, the principal result of this paper is the following one:
i
Theorem 1.3. Let M be a closed, orientable and irreducible 3-manifold and let Σ=Σ (cid:116)...(cid:116)Σ
1 l
be an embedded link in M. Suppose the existence of a sequence M of hyperbolic cone-manifolds
i
with topological type (M,Σ) and having cone angles α ∈(0,2π]. If
ij
(1.1) sup{L (Σ ) ; i∈N and j ∈{1,...,l}}<∞
Mi j
and the sequence M collapses, then M is Seifert fibered or a Sol manifold.
i
As a consequence of the theorem (1.3), we obtain the following result related to the Thurston’s
question (1.1).
Corollary 1.4. Let M be a closed and orientable hyperbolic 3-manifold and suppose the existence
of a finite union of simple closed geodesics Σ in M. Let M be a (angle decreasing) deformation of
α
this structure along a continuous path of hyperbolic cone-structures with topological type (M,Σ)
and having the same cone angle α∈(θ,2π]⊂[0,2π] for all components of Σ. If
(1.2) sup{L (Σ ) ; α∈(θ,2π] and j ∈{1,...,l}}<∞,
Mα j
DEFORMATIONS OF HYPERBOLIC CONE-STRUCTURES 3
then every convergent (in the Hausdorff-Gromov sense) sequence M , with α converging to θ,
αi i
does not collapses.
If the deformation in the statement of the corollary is supposed to be maximal, we remind that
θ =0 if and only if θ ≤π. This is a consequence of Kojima’s work in [Koj].
2. Metric Geometry
Given a metric space Z, the metric on Z will always be denoted by d (·,·). The open ball
Z
of radius r >0 about a subset A of Z is going to be denoted by
(cid:83)
B (A,r)= {z ∈Z ; d (z,a)<r}.
Z Z
a∈A
A metric space Z is called a length space (and its metric is called intrinsic) when the distance
between every pair of points in Z is given by the infimum of the lengths of all rectificable curves
connecting them. When a minimizing geodesic between every pair of points exists, we say that Z
is complete.
For all k ∈ R, denote M2 the complete and simply connected two dimensional Riemannian
k
manifold of constant sectional curvature equal to k. Given a triple of points (x; y,z) of Z, a
comparison triangle for the triple is nothing but a geodesic triangle ∆ (x,y,z) in M2 with vertices
k k
x, y and z such that
d (x,y)=d (x,y) , d (y,z)=d (y,z) and d (z,x)=d (z,x).
M2 Z M2 Z M2 Z
k k k
Note that a comparison triangle always exists when k ≤ 0. The k-angle of the triple (x; y,z) is,
by definition, the angle (cid:93) (x; y,z) of a comparison triangle ∆ (x,y,z) at the vertex x (assuming
k k
the triangle exists).
Definition2.1. Afinitedimensional(intheHausdorffsense)lengthspaceZiscalledanAlexandrov
space of curvature not smaller than k ∈ R if every point has a neighborhood U such that, for all
pointsx,y,z ∈U,theangles(cid:93) (x; y,z),(cid:93) (y ; x,z)and(cid:93) (z ; x,y)arewelldefinedandsatisfy
k k k
(cid:93) (x; y,z)+(cid:93) (y ; x,z)+(cid:93) (z ; x,y)≤2π.
k k k
Wepointoutthateveryhyperboliccone-manifoldisanAlexandrovspaceofcurvaturenotsmaller
than −1.
Suppose from now on that Z is a n dimensional Alexandrov space of curvature not smaller than
k ∈ R. Consider z ∈ Z and λ ∈ (0,π). The point z is said to be λ-strained if there exists a set
{(a ,b )∈Z×Z ; i∈{1,...,n}}, called a λ-strainer at z, such that (cid:93) (x; a ,b )>π−λ and
i i k i i
(cid:110)(cid:12) π(cid:12) (cid:12) π(cid:12) (cid:12) π(cid:12)(cid:111)
max (cid:12)(cid:93) (x; a ,a )− (cid:12), (cid:12)(cid:93) (x; b ,b )− (cid:12),(cid:12)(cid:93) (x; a ,b )− (cid:12) <λ
(cid:12) k i j 2(cid:12) (cid:12) k i j 2(cid:12) (cid:12) k i j 2(cid:12)
for all i (cid:54)= j ∈ {1,...,n}. The set R (Z) of λ-strained points of Z is called the set of λ-regular
δ
points of Z. It is a remarkable fact that R (Z) is an open and dense subset of Z.
δ
Recall now, the notion of (pointed) Hausdorff-Gromov convergence (see [BBI]):
Definition 2.2. Let (Z ,z ) be a sequence of pointed metric spaces. We say that the sequence
i i
(Z ,z ) converges in the (pointed) Hausdorff-Gromov sense to a pointed metric space (Z,z ), if the
i i 0
following holds: For every r >ε>0, there exist i ∈N and a sequence of (may be non continuous)
0
maps f :B (z ,r)→Z (i>i ) such that
i Zi i 0
i. f (z )=z ,
i i 0
ii. sup{d (f (z ),f (z ))−d (z ,z ) ; z ,z ∈Z}<ε,
Z(cid:48) i 1 i 2 Z 1 2 1 2
4 ALEXANDREPAIVABARRETO
iii. B (z ,r−ε)⊂B (f (B (z ,r)),ε),
Z 0 Z i Zi i
iv. f (B (z ,r))⊂B (z ,r+ε).
i Zi i Z 0
Its a fundamental fact that the class of Alexandrov spaces of curvature not smaller than k ∈
R is pre-compact with respect to the notion of convergence in the Hausdorff-Gromov sense. In
particular, every pointed sequence of hyperbolic cone-manifolds with constant topological type has
a subsequence converging (in the Hausdorff-Gromov sense) to a pointed Alexandrov space which
may have dimension strictly smaller than 3, although its curvature remains bounded from below
by −1.
The Fibration Theorem is an useful tool for the study of sequences of Alexandrov spaces. The
first version of this theorem, due to Fukaya (see [Fuk]), concerns the Riemannian manifolds with
pinched curvature. Later this theorem was extended by Takao Yamaguchi to Alexandrov Spaces
(see[Yam],[SY]and[Bel]). Fortheconvenienceofthereader,wepresentheretheprecisestatement
of the Yamaguchi theorem that is going to be needed in this work (see [Bar]):
Theorem 2.3 (Yamaguchi). Let (M ,p ) be a sequence of pointed complete Riemannian mani-
i i
folds of dimension 3 with sectional curvature bounded from below by −1. Suppose that the sequence
(M ,p ) converges in the Hausdorff-Gromov sense to a pointed complete length space (Z,z) of di-
i i
mension2(resp. dimension1), thenthereexistsaconstantλ>0satisfyingthefollowingcondition:
For every compact domain Y of Z contained in R (Z) and for sufficiently large i = i (Y) ∈ N ,
λ 0 0
there exist
• a sequence τ >0 (i>i ) converging to zero,
i o
• a sequence N (i>i ) of compact 3-submanifolds of M (perhaps with boundary)
i o i
• a sequence of τ -approximations p:N →Y which induces a structure of locally trivial fibre
i i
bundle on N . Furthermore, the fibers of this fibration are circles (resp. spheres, tori).
i
3. Sequences of Hyperbolic cone-manifolds
RecallthatM denotesaclosed,orientableandirreducibledifferentialmanifoldofdimension
3andthatΣ=Σ (cid:116)...(cid:116)Σ denotesanembeddedlinkinM. Asequenceofhyperboliccone-manifolds
1 l
with topological type (M,Σ) will always be denoted by M .
i
Given a sequence M as above, fix indices i∈N and j ∈{1,...,l}. For sufficiently small radius
i
R>0, the metric neighborhood
B (Σ ,R)={x∈M ; d (x,Σ )<R}
Mi j i Mi j
of Σ is a solid torus embedded in M . The supremum of the radius R > 0 satisfying the above
i
propertywillbecallednormalinjectivityradiusof Σ in M anditisgoingtobedenotedbyR (Σ ).
j i i j
Analogously we can define R (Σ), the normal injectivity radius of Σ. It is a remarkable fact (see
i
[Fuj] and [HK]) that the existence of a uniform lower bound for R (Σ) ensures the existence of a
i
sequence of points p ∈ M such that the sequence (M ,p ) converges in the Hausdorff-Gromov
ik ik ik
sense to a pointed hyperbolic cone-manifold (M ,p ) with topological type (M,Σ). Moreover,
∞ ∞
M must be compact provides that cone angles of M are uniformly bounded from below.
∞ ik
ThepurposeofthissectionistoprovetheTheorem(1.3). Thissectionisdividedintotwoparts.
The first part contains some preliminary results whereas the remaining one presents the proof of
the theorem.
Let us point out that, throughout the rest of the paper, the term ”component” is going to stand
for ”connected component”
DEFORMATIONS OF HYPERBOLIC CONE-STRUCTURES 5
3.1. Preliminary results. Let us begin with three elementary lemmas which will be important
for the proof of Theorem (1.3).
Lemma 3.1. Suppose that M −Σ is hyperbolic and let T be a two dimensional torus embedded in
M −Σ. Then T separates M. Moreover, one and only one of the following statements holds:
i. T is parallel to a component of Σ (hence it bounds a solid torus in M),
ii. T is not parallel to a component of Σ and it bounds a solid torus in M −Σ,
iii. T isnotparalleltoacomponentofΣanditiscontainedinaballB ofM−Σ. Furthermore,
T bounds a region in B which is homeomorphic to the exterior of a knot in S3.
Proof. Suppose that T is not parallel to a connected component of Σ. Since M −Σ is hyperbolic,
T is compressible in M −Σ and hence it splits M into two components, say U and V. Denote by
Σ and by Σ the subsets of Σ contained respectively in U and in V. Without loss of generality
U V
suppose Σ (cid:54)=∅.
U
Let D be a compression disk for T. Since M−Σ is irreducible and Σ (cid:54)=∅, the sphere obtained
U
bythecompressionofT alongDboundsaballB ⊂M−ΣintheoppositesideofΣ . Inparticular,
U
this implies that Σ = ∅, since Σ and Σ lie in different sides of S, by construction. Thus V
V U V
must satisfy one of the assertions (ii) or (iii) according to whether D ⊂V or D ⊂U. (cid:3)
Lemma 3.2. Given a closed ball B ⊂M −Σ and an embedded two dimensional torus T ⊂B, let
U and W be the compact three manifolds (with toral boundary) obtained by surgery of M along T
(suppose W ⊂ B). If W is homeomorphic to the exterior of a knot in S3, then we can replace W
by a solid torus V without changing the topological type of M.
Proof. Let B(cid:48) be a closed ball and h : ∂B → ∂B(cid:48) a diffeomorphism such that B ∪ B(cid:48), the
h
manifold obtained by gluing B and B(cid:48) through h, is diffeomorphic to S3. Note that the manifold
V(cid:48) = (U ∩B)∪ B(cid:48) is a closed solid torus. Let V be a closed solid torus and g : ∂V → ∂V(cid:48)
h
a diffeomorphism such that V ∪ V(cid:48), the manifold obtained by gluing V and V(cid:48) through g, is
g
diffeomorphic to S3. Thus the compact three manifold B =(V(cid:48)∪ V)−int(B(cid:48))=(U ∩B)∪ V is
g g
diffeomorphic to a closed ball.
It is a standard result of low dimensional topology that the manifold
(M −int(B))∪ B =(M −W)∪ V
Id g
obtained by using Id:∂B →∂B as gluing map, is diffeomorphic to M. (cid:3)
Lemma 3.3. Consider ε ∈ (0,1) and let M be a hyperbolic cone-manifold with topological type
(M,Σ) together with a metric neighborhood V of Σ whose components are homeomorphic to solid
tori. Denote by g the hyperbolic metric on M−Σ and let h be a Riemannian metric on M such
that:
i. h coincides with g on M−V,
ii. h is a ε-perturbation of g on M −Σ, i.e.
(cid:12) (cid:12)
(3.1) (cid:12)(cid:107)v(cid:107) −(cid:107)v(cid:107) (cid:12)<ε,
(cid:12) g h(cid:12)
(cid:110) (cid:111)
for every x∈M−Σ and v ∈K = u∈T (M−Σ) ; (cid:107)u(cid:107) ≤2 .
x x g
Then the Hausdorff-Gromov distance between M and the Riemannian manifold N = (M,h) is
smaller than 4εD, where D =1+diam (M).
M
6 ALEXANDREPAIVABARRETO
Proof. Toprovethelemma,itsufficestocheckthattheidentitymapId:M→N isa2εD-isometry.
Fix x,y ∈ M. Since Σ has codimension 2, there exists a differentiable curve α : [0,L] → M
(parametrized by arc length in M) such that α((0,L))⊂M−Σ and L<d (x,y)+ ε. Then
M 2
ε
d (x,y)−d (x,y) ≤L (α)−L (α)+
N M N M 2
(cid:90) L(cid:12) (cid:12) ε
≤ (cid:12)(cid:107)α(cid:48)(t)(cid:107) −(cid:107)α(cid:48)(t)(cid:107) (cid:12) dt +
(cid:12) g h(cid:12) 2
0
ε2 ε
<ε.d (x,y)+ + ≤εD
M 2 2
and, in particular, diam (M)≤2D.
N
Similarly, there exists a differentiable curve β : [0,L(cid:48)] → N (parametrized by arc length
in N) such that β((0,L(cid:48))) ⊂ N − Σ and L(cid:48) < d (x,y) + ε. Inequality (3.1) implies that
N 2
{u∈T (M−Σ) ; h (u,u)=1}⊂K , for all z ∈N −Σ. So
z z z
(cid:12) (cid:12)
(cid:12)(cid:107)β(cid:48)(t)(cid:107) −(cid:107)β(cid:48)(t)(cid:107) (cid:12)<ε,
(cid:12) g h(cid:12)
for all t ∈ [0,L(cid:48)]. Analogously to the preceding case it then follows that d (x,y)−d (x,y) ≤
M N
2εD. (cid:3)
3.2. Proof of the Theorem (1.3). The purpose of this section is to study a collapsing
sequence M . According to [Fuj, theorem 1], the sequence M cannot collapse when lim R (Σ )=
i i i j
i→∞
∞, for all components Σ of Σ. Given p ∈ Σ , we can assume without loss of generality that
j 1
sup{R (Σ ) ; i∈N}<∞andthatthesequence(M ,p)convergesintheHausdorff-Gromovsense
i 1 i
to a pointed Alexandrov space (Z,z ). We are interested in the case where the length of the
0
singularity remains uniformly bounded, i.e. where
sup{L (Σ ) ; i∈N,j ∈{1,...,l}}<∞.
Mi j
AgainanAscoli-typeargumentimplies(passingtoasubsequenceifnecessary)thateachcomponent
Σ of Σ satisfies one, and only one, of the following statements:
j
(1) sup{d (p,Σ ) ; i∈N}<∞andΣ convergesintheHausdorff-Gromovsensetoaclosed
Mi j j
curve ΣZ ⊂Z,
j
(2) limd (p,Σ )=∞.
i∈N Mi j
This dichotomy allows us to write Σ=Σ (cid:116)Σ , where Σ contains the components Σ of Σ which
0 ∞ 0 j
satisfies item (1) (in particular Σ ⊂ Σ ) and Σ those that satisfies the item (2). Now consider
1 0 ∞
the compact set
Σ = (cid:83) ΣZ ⊂Z.
Z j
Σj⊂Σ0
The following proposition provides information on the dimension of the limit of a sequence of
hyperbolic cone-manifolds which collapses.
Lemma 3.4. Given a sequence p ∈ M, suppose that the sequence (M ,p ) converges in the
i i i
Hausdorff-Gromov sense to a pointed Alexandrov space (Z,z ). If the sequence M collapses and
0 i
verifies
sup{L (Σ ) ; i∈N, j ∈{1,...,l}}<∞,
Mi j
then the dimension of Z is strictly smaller than 3.
DEFORMATIONS OF HYPERBOLIC CONE-STRUCTURES 7
Proof. We can assume that there exists z ∈Z−Σ , since the equality Σ =Z can only happens
∞ Z Z
when the dimension of Z is strictly smaller than 3. Set δ = d (z ,Σ ) > 0 and take a sequence
Z ∞ Z
q ∈M −Σ such that
i
(cid:18) (cid:19) (cid:18) (cid:19)
δ δ
lim q =z and B q , ∩B Σ, =∅.
i→∞ i ∞ Mi i 3 Mi 3
By definition, the sequence (M ,q ) converges in the Hausdorff-Gromov sense to (Z,z ). Since the
i i ∞
sequence M collapses, we have lim rMi (q )=0.
i inj i
i→∞
According to [HK, Lemma 3.8] we can change the metrics of M on B (cid:0)Σ,δ(cid:1)−Σ to obtain a
i Mi 3
sequence of complete Riemannian manifolds N homeomorphic to M −Σ which have finite volume
i
andpinchedsectionalcurvature. Withoutlossofgenerality,wecanassumethatthesequence(N ,q )
i i
converges in the Hausdorff-Gromov sense to a pointed Alexandrov space (Y,y ). Note that, by
∞
construction, the balls B (cid:0)q ,δ(cid:1) and B (cid:0)q ,δ(cid:1) are isometric. Then the balls B (cid:0)y ,δ(cid:1) and
Ni i 3 Mi i 3 Y ∞ 3
B (cid:0)z ,δ(cid:1) are also isometric and we have lim rNi (q )=0.
Z ∞ 3 i→∞ inj i
According to [Gro, Corollary 8.39], B (cid:0)z ,δ(cid:1) has Hausdorff dimension strictly smaller than 3.
Z ∞ 3
Given that the dimension of an Alexandrov space is a well-defined integer, it follows that Z also
has dimension strictly smaller than 3. (cid:3)
The preceding lemma says that the only possibilities for the dimension of Z are 2, 1 or 0. The
proof of Theorem (1.3) will accordingly be split in three cases.
3.2.1. Case where Z has dimension 2. Itisaclassicalresultofmetricgeometrythatalltwodimen-
sionalAlexandrovspacesaretopologicalmanifoldsofdimension2,possiblywithboundary. Wecan
describe the boundary of Z as
(cid:70)
∂Z = ∂Z
k
k∈Γ
where ∂Z are the connected components of ∂Z. Note that each component ∂Z is homeomorphic
k k
to a circle or to a straight line.
Theorem 3.5 (collapsing-dimension2). Suppose the existence of a point p∈Σ such that the se-
1
quence (M ,p) converges (in the Hausdorff-Gromov sense) to a two dimensional pointed Alexandrov
i
space (Z,z ) and also that sup{R (Σ ) ; i∈N}<∞. Assuming that
0 i 1
sup{L (Σ ) ; i∈N, j ∈{1,...,l}}<∞,
Mi j
it follows that:
i. M is Seifert fibered,
ii. if ∂Z (cid:54)=∅, then ∂Z has only one connected component and M is a lens space,
iii. if Z is not compact, then limR (Σ )=∞, for any component Σ of Σ .
i j j ∞
i∈N
Proof. Accordingto[Sal],themetricofhyperboliccone-manifoldsM insomesmallneighborhoods
i
of Σ can be deformed to yield a sequence of complete Riemannian manifolds N (homeomorphic
i
to M) with sectional curvature not smaller than −1. Furthermore thanks to Lemma (3.3), we can
suppose that the sequence (N ,p) converges (in the Hausdorff-Gromov sense) to (Z,z ).
i 0
If Z is compact, the statement follows directly from the results of Shioya and Yamaguchi in
[SY]. To be more precise, when Z does not have boundary, [SY, Theorem 0.2] applied to the
sequence N gives that M is Seifert fibered. When Z has boundary, [SY, Corollary 0.4] applied
i
to the sequence N implies (M is irreducible) that Z is a closed disk with at most one cone in its
i
8 ALEXANDREPAIVABARRETO
interior. Furthermore, [SY, Theorem 0.3] implies that M is a lens space (in particular it is also
Seifert fibered).
Suppose from now on that Z is not compact. Since the normal injectivity radius of components
of Σ are uniformly bounded, there exists R>0 such that
0
B (Σ ,R (Σ ))⊂B (p,R/2)
Mi j i j Mi
foralli∈NandforeverycomponentΣ ⊂Σ . LetK beacompactandconnectedtwodimensional
j 0
submanifold of Z such that B (z ,R) ⊂ K (therefore Σ ⊂ K), ∂K is a disjoint union of circles
Z 0 Z
and Z−K is a disjoint union of components of infinite diameter.
Set Λ={k ∈Γ; ∂Z ∩K (cid:54)=∅}. If ∂Z (cid:54)=∅, we assume also that:
k
• Λ(cid:54)=∅, that is K∩∂Z (cid:54)=∅,
• ∂Z ∩K =∂Z , for all k ∈Λ such that the component ∂Z is compact,
k k k
• ∂Z ∩K is connected, for all k ∈Λ.
k
For each k ∈ Λ, denote by ∂K the (unique by construction) connected component of ∂K such
k
that ∂Z ∩∂K (cid:54)=∅. The boundary of K is given by
k k
(cid:70) (cid:70)
∂K = ∂K (cid:116) ∂K ,
k m
k∈Λ m∈Γ−Λ
where ∂K are the components of ∂K which does not intersect the boundary of Z.
m
Let λ be the constant given by the Fibration Theorem (2.3). Since K is compact, the sets of
λ-cone-points
C ={z ∈K−∂Z ; L(T Z)≤2π−λ}
z
C ={z ∈∂Z ∩K ; L(T Z)≤π−λ}⊂∂K (k ∈Λ)
k k z k
are finite (cf. [SY, Theorem 2.1]). Modulo increasing K, we can assume that C is contained in its
interior. Let s (cid:29)s >0 be such that
1 2
• B [z,s ]⊂int(K) and B [z,s ] is homeomorphic to D2, for all z ∈C,
K 1 K 1
• B [z,s ]∩B [z(cid:48),s ]=∅, for all z,z(cid:48) ∈C∪ (cid:83) C ,
K 1 K 1 k
k∈Λ
(cid:83)
• For all z ∈C∪ C and for all k ∈Λ, we have
k
k∈Λ
B (∂K ,s )∩B (z,s )(cid:54)=∅ if and only if z ∈C .
k k 2 K 1 k
Consider the compact subset Y of Z defined by
(cid:83) (cid:70)
Y =K− B (z,s )− U ⊂R (Z),
K 1 k λ
z∈C k∈Λ
(cid:83)
where U =B (∂Z ,s )∪ B (z,s ), for every k ∈Λ. The boundary of Y is given by
k K k 2 K 1
z∈Ck
(cid:70) (cid:70) (cid:70)
(3.2) ∂Y = ∂Y (cid:116) ∂Y (cid:116) ∂K ,
k z m
k∈Λ z∈C m∈Γ−Λ
where ∂Y and ∂Y are the components of ∂Y which intersect respectively the regions U and
k z k
B (z,s ).
K 1
Without loss of generality, the Fibration Theorem (2.3) gives us a sequence N of compact and
i
connected three dimensional submanifolds of M, a sequence τ > 0 that converges to zero and
i
a sequence of τ -approximations p : N → Y which induces a structure of locally trivial circle
i i i
bundle on N . Since the manifold M is orientable, it follows that the connected components of
i
∂N =p−1(∂Y) are two dimensional tori.
i i
DEFORMATIONS OF HYPERBOLIC CONE-STRUCTURES 9
Let C be a component of ∂Y and T the component of ∂N associated with it (T = p−1(C)).
i i i i
By construction, we can choose base points q ∈ T and q ∈ C such that the pointed components
i i
(B ,q )ofN −int(N )associatedwiththetorusT (∂B =T ),convergesintheHausdorff-Gromov
i i i i i i i
sense to the pointed component (B,q) of Z−int(Y) associated to C (∂B =C).
Figure 1.
The above paragraph shows that the bijection between the boundary components of Z−Y and
those of N −N induces a bijection between the components of Z −Y and those of N −N .
i i i i
Moreover, the diameter of the components of N −N associated with a component B of Z −Y
i i
goes to infinity if and only if B is noncompact.
Without loss of generality, we can suppose that Σ ⊂ N −N , for all i ∈ N. It follows from
∞ i i
lemma (3.1) that the singular components of N −N are solid tori whose souls are components of
i i
Σ . Moreover, as the boundary components of N remain a finite distance from the base point,
∞ i
thenormal injectivityradiiofthe componentsofΣ becameinfinitewith i. Notethatthis implies
∞
thatthesingularcomponentsofN −N mustbeassociatedwithnoncompactcomponentsofZ−Y.
i i
Lemma 3.6. There exists i ∈ N such that, for every i > i , the components of N −N are
0 0 i i
homeomorphic to either a solid torus or to the exterior of a knot in S3 which, in addition, is
contained in an embedded ball in M −Σ.
Proof of Lemma 3.6 : A boundary torus of N −N has 3 possible natures according to the
i
decomposition (3.2) of ∂Y. We shall consider separately each of there possibilities.
1st type : The torus Tz =p−1(∂Y )⊂∂N , z ∈C.
i i z i
Consider a point z ∈ C. By rescaling the metric with respect to the length of the fibers and
using the stability theorem of Perelman (cf. [Kap]) , Shioya and Yamaguchi show in [SY, theorem
0.2] that the region Bz of N −N bounded by the torus Tz is a solid torus, for sufficiently large i.
i i i i
Furthermore, the solid torus Bz is glued on N without ”killing” the fiber.
i i
10 ALEXANDREPAIVABARRETO
Figure 2. Tori of second type
2nd type: The torus Tk =p−1(∂Y )⊂∂N , k ∈Λ.
i i k i
Given an index k ∈ Λ, we can decompose the component ∂Y in two simple arcs α and β with
k
the same ends, say P and Q, and such that α= U ∩∂Y . Note that α=∂Y and β is reduced to
k k k
a single point (in particular P =Q) when ∂Z is compact.
k
Let Bk be the region of N −N bounded by the torus Tk. Using the same techniques, [SY,
i i i i
Theorem 0.3] shows that we can choose, for sufficiently large i, a region Bkα inside Bk which is
i i
homeomorphic to α×D2. Moreover:
• the sequence formed by the regions Bkα converges in the Hausdorff-Gromov sense to the
i
region U ,
k
• thecircle{z}×∂D2 isafiberoftheSeifertfibrationontheboundaryofN ,foreveryz ∈α.
i
Then, for sufficiently large i, the regions Bk are homeomorphic to solid tori provided that ∂Z is
i k
compact. Suppose now that ∂Z is not compact.
k
Let DP and DQ be the disks of Bkα which are glued respectively on the fibers p−1(P) and
i i i i
p−1(Q). The fibers above β together with the disks DP and DQ yield an embedded sphere in
i i i
N −Σ. This sphere splits N into two regions: the region Bkβ = Bk −int(cid:0)Bkα(cid:1) and the region
i i i i i
(cid:16) (cid:17)
N −int Bkβ which contain the singular set Σ (cid:54)= ∅. Since the manifold N −Σ is irreducible,
i i 0 i
the region Bkβ must be homeomorphic to a ball and therefore, for sufficiently large i, the region
i
Bk =Bkα∪Bkβ is homeomorphic to a solid torus.
i i i
3th type: The torus Tm =p−1(∂K )⊂∂N , m∈Γ−Λ.
i i m i
Given an index m ∈ Γ−Λ, let Bm be the region of N −N bounded by the torus Tk. Since
i i i i
B (z ,R) ⊂ K, the torus Tm is contained in M −Σ and cannot be parallel to a component of
Z 0 i
Σ . According to Lemma (3.1), the region Bm is homeomorphic either to a solid torus (possibly
0 i
having a component of Σ as its soul) or to the exterior of a knot in S3 which is contained in an
∞
embedded ball in M −Σ. (cid:5)