The Teichmüller space of the Hirsch foliation
We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.
- 1.1 The Hirsch foliation
- 1.2 Hyperbolic metrics and Teichmüller space
- 1.3 General Hirsch foliations
- 1.4 Massage of an annulus
2 Hyperbolic metrics on planar pairs of pants
- 2.1 Hyperbolic metrics and conformal structures
- 2.2 Diffeomorphisms between hyperbolic metrics
- 2.3 Boundary admissible hyperbolic metrics
- 2.4 Admissible and symmetric sections of Teichmüller space
- 2.5 Homotopy to an admissible and symmetric section
- 3 Model hyperbolic metrics on the Hirsch foliation
- 4 Deforming hyperbolic metrics to model metrics
- 5 Contractibility of the space of foliation diffeomorphisms
The Teichmüller space of compact surfaces is deeply related to the structure of the group of self-diffeomorphims of such surfaces. For example, in [EE69] and [ES70], it is shown that the identity component of the group of self-diffeomorphisms of a compact hyperbolic surface is contractible. The proof proceeds in three steps. First, one shows that the Teichmüller space is contractible. From this one obtains that the space of hyperbolic metrics is a trivial principal fiber bundle over the Teichmüller space whose structure group is the identity component of the diffeomorphism group. Finally, since the space of hyperbolic metrics is also contractible (as can be seen via identification with Beltrami coefficients), one obtains that the fiber must be as well.
In this work we extend the above line of reasoning to the Hirsch foliation which is a well known foliation of a closed 3-manifold by non-compact, non-simply connected, hyperbolic surfaces. We prove the following results.
The Teichmüller space of the Hirsch foliation is homeomorphic to the space of continuous closed curves in the open half-plane.
The space of hyperbolic metrics on the Hirsch foliation endowed with the projection to is a trivial principal bundle.
The arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.
Theorem A provides an infinite dimensional analogue of Fenchel-Nielsen coordinates for hyperbolic metrics for the Hirsch foliations. The two functions and on the unit circle parametrizing each equivalence class of hyperbolic metrics can be interpreted as length and twist parameters associated to a certain family of disjoint closed geodesics on the leaves of the foliation.
We note that the Teichmüller theory of non-compact surfaces encounters several technical difficulties. In particular, for surfaces of infinite topological type, Teichmüller space can be defined, using either pants decompositions, complex structures, or length-spectrum. But these definitions may yield different spaces as show in [ALP11, ALPS12b] and [ALPS12a].
Another way to generalize the concept of closed surface is to consider laminations of compact spaces by surfaces. See [Ghy99] for a discussion of the extent to which the basic theorems for Riemann surfaces, such as the Uniformization, Gauss-Bonnet, or the existence of meromorphic functions, can be extended to this general foliated context.
The problem of simultaneous uniformization of the leaves of a compact lamination is of special interest in the present work: is it possible to uniformize each leaf simultaneously? And, do the uniformizations of the leaves vary continuously with the transverse parameter? In his thesis [Can93], Candel answers both of these questions affirmatively in the case where the leaves are of hyperbolic type, by constructing families of hyperbolic metrics on the leaves which vary continuously with the transverse parameter.
Teichmüller spaces for hyperbolic surface laminations (where the total space of the lamination is compact even though the leaves may not be) were first introduced by Sullivan in [Sul92] and later on in more detail in [Sul93]. They seem to be more amenable to study than the Teichmüller spaces of general non-compact surfaces. For example, as in the case of compact surfaces one can define the Teichmüller space of a lamination either in terms of complex structures or hyperbolic metrics and both definitions are equivalent (see [MS06, pg. 232]). Related to this, Candel’s version of the uniformization theorem (see [Can93]) establishes that there is a unique hyperbolic structure conformal to each Riemannian structure on any lamination by hyperbolic leaves.
In spite of these results there are only two cases prior to this work in which one understands the Teichmüller space of a concrete surface lamination in any depth.
The first case is a certain family of laminations which can be associated to the expanding maps () on the unit circle. One can show that the Teichmüller space of such a lamination is in bijection with the conjugacy classes of expanding maps of the same degree, see [Sul93] and [Ghy99].
The second family of laminations are the so-called solenoids obtained as the inverse limit of the space of finite coverings of a closed hyperbolic surface (these are a special type of the solenoidal manifolds discussed in [Sul14] and [Ver14]). In this case, certain natural problems on finite coverings such as the Ehrenpreis conjecture, can be rephrased in terms of the Teichmüller space of the corresponding lamination (see [PŠ08] and [Šar09]).
Besides Candel’s work there is, as far as the authors are aware, only one general result on Teichmüller spaces of hyperbolic surface laminations. This is the fact, established by Deroin in [Der07], that the Teichmüller space of such a lamination containing a simply connected leaf is always infinite dimensional.
In this work we investigate the Teichmüller theory of the Hirsch foliation. We recall that the Hirsch foliation was introduced in [Hir75] as an example of a foliation which is stable under perturbations of its tangent field, and has an exceptional minimal set (i.e. a compact closed set which is a union of more than one leaf and is not the entire ambient manifold). We will study a variant (also considered by Ghys for example in [Ghy95]) of his construction which is minimal (the difference with Hirsch’s construction amounts to a different choice of degree map of the circle, we will use while he used a mapping with a single attracting periodic orbit). The total space of the foliation can be defined as the orbit space of the wandering set of the domain of attraction of a solenoid mapping of the solid torus, and the foliation itself corresponds to the projection of the stable foliation of the attractor. This actually describes a family of foliations which appear for example in the study of complex Hénon maps (see [HOV94]). Our arguments apply to all of them and we will spend some time in the first section describing the different Hirsch foliations explicitly.
Hirsch foliations have also been used to produce examples of minimal foliations which are not uniquely ergodic (in the sense that they admit more than one harmonic measure, see [DV11]).
We believe that Hirsch foliations are of interest for the theory of Teichmüller spaces of hyperbolic surface laminations for the following reasons: First, the previous laminations for which Teichmüller spaces are known all contain simply connected leaves and none of them contain complicated leaves (e.g. leaves whose fundamental group is infinitely generated). Second, Sullivan solenoids are transversally Cantor and it is therefore easier to “globalize” local constructions on them. Third, minimal surface laminations with an essential holonomy-free loop are Hirsch-like in a precise sense, e.g. all leaves are obtained by pasting together elements from the same finite set of compact surfaces with boundary (see [ADMV14, Theorem 2]). Fourth, since the ambient space of the Hirsch foliation is a closed 3-manifold there may be deeper links between its Teichmüller theory and the dynamics and geometry of the manifold.
The solenoidal endomorphisms of used to define the Hirsch foliation are known to be structurally stable when extended to where is the Riemann sphere, see [IPR10]. Hence the conjugacy class of any such map contains a neighborhood of the map (see also [HOV94, Section 3] where the restrictions of the endomorphisms to a solid torus where they are injective are considered). In view of Sullivan’s result relating conjugacy classes of expanding maps of the circle and the Teichmüller of a suitable lamination one might ask the following:
What can be said about the conjugacy classes of solenoid mappings? In particular, is there a natural way of associating each such class to an element of the Teichmüller space of the associated Hirsch foliation?
The main technical issue which was solved in order to construct hyperbolic metrics on the Hirsch foliation was obtaining a global continuous section of the Teichmüller space of a pair of pants with several specific properties. For proving that all metrics on the foliation are equivalent to a “model metric” we needed on the one hand a procedure for deforming metrics on the pair pants to metrics with the aforementioned special properties, and on the other hand a procedure for deforming a general metric on the Hirsch foliation so that it admits a specific set of closed curves as geodesics. The most important tools we have used are geometric flows, in particular we use the flow on circle diffeomorphisms defined by Schwartz in [Sch92], and the curve shortening flow on a hyperbolic surface (see [Gra89]); as well as standard tools from Teichmüller theory such as the Beltrami equation (see for example [AB60]).
The Hirsch foliation we work with are constructed from a solenoidal mapping of degree . It seems possible to generalize our construction of Teichmüller space to solenoidal mapping of degree with the additional condition that they are unbraided. Following [HOV94], this means that it sends a solid torus inside itself as a (unknotted) torus. in other words, the fundamental domain is diffeomorphic to the suspension of a -connected plane domain by a rotation of angle .
We now review the structure of this paper.
In Section 1 we construct the Hirsch foliation we will be working on and its Teichmüller space. We also classify the different non-equivalent Hirsch foliations which arise from this type of construction. Finally, we introduce a technique for deforming a hyperbolic metric defined on a neighborhood of a circle in the plane using a smooth isotopy of the identity in order to make it conformal and rotationally symmetric.
In Section 2 we construct a global continuous section of the Teichmüller space of a planar pair of pants with several special properties. The first of which is that the metrics in the section are conformal and rotationally symmetric around each boundary component. Also, if one exchanges the length parameters for two “legs” of the pair of pants then the corresponding metrics given by the section differ by the 180º rotation (in particular if the two lengths are equal then the rotation is an isometry for the metric). These properties are important in order to construct metrics on the Hirsch foliation which glue together smoothly under the identifications defining the foliation.
In Section 3 we use the preceeding global section to construct a family of hyperbolic metrics on the Hirsch foliation parametrized a length parameter and a twist parameter . We also show that no two such metrics are Teichmüller equivalent.
In Section 4 we prove that any hyperbolic metric on the foliation can be deformed (using a leaf-preserving isotopy which is leafwise smooth) to one of the model metrics. This is done in two cases. In the first one we assume that the given metric already has a certain distinguished family of curves as geodesics so one can operate separately on each pair of pants. We then show how to deform a general metric to this case using the curve-shortening flow.
1.1 The Hirsch foliation
Consider the smooth endomorphism of (where ) defined by
Let (where is the open unit disk). The closed solid torus is mapped diffeomorphically into by . In fact, restricted to is the well known solenoid map and the compact set is a hyperbolic attractor which is locally homeomorphic to times a Cantor set.
The Hirsch foliation.
Let be the union of preimages of . The quotient (where two points are equivalent if they belong to the same complete orbit, i.e. if for some ) is a compact boundaryless smooth manifold and the foliation of by leaves of the form descends to . The resulting foliated compact boundaryless -manifold (where both the manifold and the leaves of the foliation are smooth) is what we will call from now on the Hirsch foliation .
Topology of the leaves.
One can verify that each complete -orbit intersects the set (which is a -manifold with two boundary components which are -dimensional tori) at exactly one interior point or at one point on each boundary component: is a fundamental domain of . Hence is obtained by pasting the two boundaries of using . The sets of the form are pairs of pants, and the partition of into these pants when pasted using yields the Hirsch foliation. With this description it is simple to see that the leaves of the Hirsch foliation are homeomorphic to either the two-dimensional torus minus a Cantor set or the two-dimensional sphere minus a Cantor set (depending on whether the leaf contains a pair of pants such that is periodic under iterated squaring or not).
1.2 Hyperbolic metrics and Teichmüller space
By a hyperbolic metric on the Hirsch foliation we mean an assignment of a hyperbolic Riemannian metric to each leaf which varies continuously transversally in local charts with respect to the topology of local smooth convergence. The space of such metrics, endowed with the topology of locally uniform smooth convergence, will be denoted by .
An identity isotopy of the Hirsch foliation (sometimes we will just say leaf isotopy, or leaf-preserving isotopy) is a continuous function such that is the identity, and is a self-diffeomorphism when restricted to any leaf. Furthermore one demands that in local foliated charts varies continuously in the smooth topology with respect to the transverse parameter.
Two hyperbolic metrics are said to be equivalent if there exists an identity isotopy such that when is restricted to any leaf it is an isometry between and . In other words the metric is the pushforward of with respect to .
The Teichmüller space of the Hirsch foliation is by definition the space of equivalence classes of Riemannian metrics under leaf-preserving identity isotopies.
In our special case these definitions can be given much more explicitly. Any hyperbolic metric on the Hirsch foliation can always be lifted to yielding a -periodic family of metrics , where is defined on .
Defining so that one may identify each with a metric defined on .
The transverse continuity of the hyperbolic metric translates as follows. If and then there exists a compact neighborhood of such that all metrics with large enough are defined in and can be written as where the functions and (the coefficients of ) converge in the smooth (i.e. ) topology to the corresponding coefficients for on .
The definition of convergence in the space of hyperbolic metrics can be similarly written in these terms. A sequence of metrics converges to a metric if and only if taking and as their lifts there exists for each a closed interval containing in its interior and a compact neighborhood of such that all metrics are defined on for all and converge to on in the smooth topology uniformly with respect to .
1.3 General Hirsch foliations
The Hirsch foliation we described before is a very concrete algebraic model. Hirsch’s original construction [Hir75] is more topological. We wish to prove here that the space of metrics we will describe below does not depend on the algebraic model we chose.
The sequel seems folklore and must be well known to the specialist. But even though the Hirsch foliation has been studied for some time now (see for example [CC00, DV11, Ghy95]), it has been difficult to locate a careful construction of the different Hirsch foliations. Hence we found useful to give a topological discussion about it. For basic -manifold theory we refer to [Hat07].
1.3.1 Seifert bundle over the pair of pants
A suspended manifold.
Here a pair of pants will be a surface with three boundary components which is diffeomorphic to the symmetric planar pair of pants .
Consider an orientation preserving diffeomorphism which:
leaves invariant one of the boundary components, which we call the outer component;
exchanges the other two boundary components, which we will call the inner components;
has a unique fixed point in denoted by ;
is of order two.
In the symmetric case, just consider the rotation of angle . Suspend this diffeomorphism to construct the following manifold:
This manifold fibers over the circle, with a -fiber. From now on we will refer to the boundary components of the -fibers of as meridians.
The manifold is a solid torus with an inner solid torus drilled out which winds around twice longitudinally while winding once meridianally. It has two boundary components which are tori, we will call the outer boundary torus and inner one respectively. The discussion below is again valid with any equivalent pant bundle over the circle which is equivalent to .
We recall that a surface embedded in a -manifold is incompressible if the morphism induced by the inclusion is injective.
The boundary components of are incompressible.
The fundamental group of is given by a semi-direct product , where denotes the free group with two generators (corresponding for example to the two inner boundary components of ), and is the morphism of induced by : it permutes the two generators of .
A meridian of represents a non trivial element of the factor (the product of the two generators). As for the factor, it can be represented by the longitud of . Hence, the inclusion induces an injection : is incompressible. The same argument provides the incompressibility of . ∎
Structure of Seifert bundle.
The suspension flow on defines a structure of Seifert bundle over with a unique exceptional fiber of type [Hat07], which corresponds to the fixed point of . We will denote this exceptional fiber by .
Remember that a -manifold is said to be irreducible if every embedded -sphere bounds a -dimensional ball. By Proposition 1.12 of [Hat07], the manifold is irreducible (it is clearly not one of the exceptions listed in this proposition).
Every diffeomorphism of is homotopic to a Seifert fiber preserving diffeomorphism.
See the classification of Seifert bundles given in Theorem 2.3 of [Hat07]. ∎
The orientations of the pair of pants and of provide a natural orientation on . With this orientation, the inner (resp. outer) Seifert fiber and the meridian provide two homology classes (resp. ) and of (resp. ), and they have intersection number (resp. ).
Let be a foliation of by pants which are transversal to the Seifert fibration. Then there exists and integer such that the outer boundary component of all pairs of pants in are in the homology class on and the inner boundaries of all such pants are the class on .
The manifold is naturally a circle bundle over the -orbifold obtained by quotienting by , which is homeomorphic to an annulus. Consider an arc in linking the boundaries, and lift it to . We obtain an annulus everywhere transverse to the fibration . The boundary components of this annulus are by definition outer and inner Seifert fibers.
Now, note that is simply connected, in such a way that the restriction of the fiber bundle to is a trivial circle fibration over the simply connected manifold with boundary, which, topologically, is a closed band. The boundary of is therefore the union of four annuli, two of which, denoted by and , are copies of .
In particular, since the base is simply connected, any foliation such as described in the statement of the lemma provides a foliation of which is isotopic along the fibers to the trivial product foliation. The gluing of and determines the type of foliations.
Now, such a gluing is determined by Dehn twists along (since any class of isotopy of diffeomorphisms of the annulus is represented by a Dehn twist). Since boundary components of are Seifert fibers, we can conclude the proof of the lemma. ∎
1.3.2 Gluing the boundary components
The Hirsch foliation.
Consider an orientation preserving diffeomorphism which sends meridians onto meridians, and consider the manifold obtained by gluing the two boundary components of using .
The manifold is foliated by pairs of pants, which induce two foliations of the boundary components by circle (the two meridian foliations). By definition, sends the first meridian foliation onto the second meridian foliation. The gluing by then provides a foliation of by surfaces that we denote by and that we call the Hirsch foliation associated to .
The circle yields naturally a circle in which is transverse to all leaves of . Hence the foliation is taut in the sense of [Cal07].
The graph manifold.
The manifold possesses a natural torus , that as we prove later, is canonical in the sense that it is the unique JSJ torus of . The resulting manifold is called a graph manifold, and is not a Seifert bundle itself.
Let be an orientation preserving diffeomorphism that preserves the meridians. Then is not a Seifert bundle.
We are going to work inside . possesses a unique structure of Seifert manifold (see Lemma 1.2), so it is enough to see that does not send the Seifert fibers of onto that of .
A Seifert fiber of intersects each meridian twice, and a Seifert fiber of intersects each meridian only once. Since sends diffeomorphically meridians of onto meridians of , it implies that cannot preserve the Seifert fibers. This concludes the proof. ∎
Let be an orientation diffeomorphism that preserves the meridians. Then is irreducible and the torus is incompressible.
Let us prove first that is incompressible. We will use the Loop Theorem [Hat07]: let be an embedded closed -disc such that is an embedded circle in . We have to prove that is null homotopic in . The interior of does not meet : there is an embedded copy of inside such that is included in one of the two boundary components or . It is enough to prove that is null homotopic inside this component. But this is true since has an incompressible boundary.
Now let us prove that irreducible. Notice that has an incompressible torus: in particular its fundamental group possesses a copy of and is not finitely covered by . Moreover it possesses a taut foliation, so by Novikov’s theorem (see Theorem 4.35 of [Cal07]), it is irreducible. ∎
The JSJ torus.
We show now that inside the manifold , the torus is canonical.
Recall that any compact and irreducible -manifold can be canonically decomposed into pieces that are either Seifert or atoroidal (any incompressible torus is isotopic to a boundary component) and acylindrical (any properly embedded annulus is isotopic, fixing the boundary, to a subannulus of a boundary component) by cutting along a collection of incompressible tori. Such a collection of tori is unique up to isotopy. This decomposition is called the JSJ decomposition, and decomposition tori are called JSJ tori (see [Hat07]).
is, up to isotopy, the only incompressible torus of .
Uniqueness of the Hirsch foliation.
Now we intend to prove that the Hirsch foliation is unique. To see this, imagine that there is another fibration by pairs of pants of , that we denote by which is everywhere transverse to the Seifert bundle and is -invariant in the following sense.
The boundaries of the pairs of pants determines two foliations of and that we call -meridians (usual meridians will also be called -meridians) which are everywhere transverse to the Seifert fibers. We say the the family is -invariant if preserves the -meridians. Note that in that case the boundary components of and that of are freely homotopic: this comes from Lemma 1.3 and from the fact that does not preserve the Seifert fibers. Then gluing and by provides another foliation on . The next lemma implies that the this new foliation is isotopic to .
Lemma 1.7 (Uniqueness of the Hirsch foliation).
The Hirsch foliation of is unique. More precisely, consider another fibration in pairs of pants transverse to the Seifert bundle, which is -invariant. Then there exists an isotopy such that for every , and which commutes with : in .
Let be a fibration of in pairs of pants everywhere transverse to the Seifert bundle, which is -invariant. As we noted before, and -meridians are freely homotopic. Hence if one lifts the fibration to , the -meridians lift as simple closed curves (they are freely homotopic to the lifts of -meridians).
By hypothesis, all the pairs of pants are everywhere transverse to the lines : may be see as a graph of a smooth function satisfying the equivariance relation .
In particular, the lifts of the outer -meridians are graphs over those of outer -meridians. Using the vertical flow and the function above, it is possible to isotope -meridians to corresponding -meridians. Pushing this isotopy by shows how to isotope the inner -meridians onto inner -meridians. These isotopies may be extended to neighborhoods of and in order to isotope to a family sharing the same properties, and coinciding moreover with near the boundary, via an isotopy which commutes with .
Using one more time that the function in enables us to glue the isotopy above with an isotopy which sends the interior of to that of . The resulting isotopy stays -invariant. ∎
1.3.3 A homological invariant
The twisting number.
Define as the intersection number, inside , of the homology classes and .
The intersection number is always odd.
First, in a natural basis of integer homology of , can be written as . In particular, it is not the power of some homology class.
Inside , we have a natural basis of the homology defined by and (their intersection number is ). Since sends meridian onto meridian (and preserves the orientation), and the representation of in this basis is . Since this is not the power of some homology class (the action of on the homology is invertible), has to be odd. ∎
A topological invariant.
Let be two meridian preserving diffeomorphisms. Then the following properties are equivalent.
and are diffeomorphic.
There exists a diffeomorphism which conjugates and .
The Hirsch foliations corersponding to and are conjugate.
First, note that the third and fourth assertion clearly imply the second one.
Assume that . Then since and send diffeomorphically meridian on meridian, we see that they induce the same action in the first homology of the -torus. Hence they have the same isotopy type, and the glued manifolds and are diffeomorphic.
Now, assume that and are diffeomorphic: denote by a diffeomorphism between them. Since is a diffeomorphism, is incompressible: it is isotopic to . After performing an isotopy, one can ask that .
This implies that induces a diffeomorphism of , still denoted by , such that the commutation relation holds in restriction to . By Lemma 1.2 is homotopic to a fiber preserving diffeomorphism. Hence, preserves the homology classes and . Since it conjugates and , we deduce that .
We want to prove that in that case, the corresponding Hirsch foliations are conjugate. The image by of the fibration provides a family of pairs of pants which is transverse to the Seifert bundle (since preserves it) and is -invariant (since conjugates the actions of and ). Lemma 1.7 provides an isotopy from to which is -invariant. In other terms, the two Hirsch foliations are conjugate.
The other implications are obvious.
This homological invariant will be referred to as the twisting number of the corresponding Hirsch foliation.
1.3.4 Algebraic models
We have already met Smale’s solenoid :
Identify and a fundamental domain of given by . Then, clearly induces a diffeomorphism (still denoted by ) from to which preserves the meridians, and satisfies .
Hence every Hirsch foliation with same twisting number is conjugated to this model, which we will study in detail in what follows.
It is easily showed that if in , we compose the diffeomorphism with a positive Dehn twist of , the twisting number is increased by . Hence these diffeomorphisms provide models of the Hirsch foliation for every odd integer . Algebraic models exist, and appear in [HOV94]. They are defined by maps given by the formula:
1.4 Massage of an annulus
The goal of this paragraph is to describe a procedure for deforming hyperbolic metrics around a geodesic circle via identity isotopy. This procedure can later be applied around each boundary component of (via the affine maps for the left and right boundaries) to construct the homotopy of Theorem 2.7. However, we will also use the procedure directly later on for deforming hyperbolic metrics on the Hirsch foliation (see the proof of Lemma 4.1).
1.4.1 Standard hyperbolic annuli
Standard hyperbolic annuli.
For each positive length there is a unique conformal metric on the annulus in which is hyperbolic, rotationally invariant, and such that the unit circle is a geodesic of length .
In order to see this consider the strip model of the hyperbolic plane. That is, consider the metric
on the strip (this is obtained from the usual upper half plane model by pullback under the conformal map ).
The metric is the pushforward of the above metric via the conformal covering map .
The couples will be referred to as the standard hyperbolic annuli.
Standard hyperbolic metrics.
Suppose that is a hyperbolic Riemannian metric defined on some region in the plane for which a Euclidean circle is a closed geodesic of length . We say is standard around if it coincides, on some neighborhood of , with the pushforward of under a conformal map of the form taking the unit circle to .
1.4.2 The Massage Lemma
Consider an annulus and denote by the set of hyperbolic metrics on with the unit circle as a geodesic. Consider the rotation of angle , i.e. . We will prove the following:
Theorem 1.10 (Massage Lemma).
For each there exists an identity isotopy such that:
Each diffeomorphism preserves the unit circle and is the identity outside the annulus defined by .
The pullback metric is standard around .
If , then we have for every , .
For each the map is continuous in the smooth topology.
The deformation will follow four steps. At each step of this procedure, we shall check that Properties 1. 2. 3. 4. are satisfied.
Deformation of the unit tangent bundle. We perform an identity isotopy for the circle to be geodesic when parametrized by Euclidean arc length. In other words, after this step the Euclidean unit tangent vector field to the circle is parallel for the metric.
Deformation of the normal bundle. We perform an identity isotopy so that the Euclidean normal vector field to is also perpendicular to for the deformed metric.
Conformality on the circle. We perform an isotopy in order to make the metric conformal to the Euclidean metric on the circle .
Standardness around the circle. We perform an isotopy to make the metric standard around . At this point (and only at this point) some standard metrics may be deformed to other equivalent standard metrics.
1.4.3 Auxiliary functions
We fix from now on a bump function, i.e. a smooth map which is on and on .
For each we will also need to fix a smooth increasing diffeomorphism which is the identity outside of the interval and has derivative at . We further suppose that is smooth with respect to both and and that that is the identity map.
1.4.4 Deformation of the tangent bundle
Parametrize the geodesic.
Until the end of the proof of the Massage Lemma (Theorem 1.10), we fix a metric . Let be the -length of the geodesic . In order to simplify the presentation, we will assume, in this paragraph only, that .
Consider an arc length parametrization of , that is an orientation preserving diffeomorphism such that where by definition is the Euclidean metric on . Such a diffeomorphism is well defined up to precomposition by a rotation.
There is a continuous section , which is characterized by the condition .
We will denote by , the group of Möbius transforms leaving invariant, by the group of rotations of , , and by , the group of translations of . These groups are conjugated by the exponential map , .
A bi-equivariant deformation retract.
We want to find a continuous path of metric to a metric for which is a geodesic when parametrized by the Euclidean length of arc. And we want these paths to be equivariant for the action of . A first step is to modify in an equivariant way the parametrization of . In order to do this, we will use a strong deformation retract from to , which is bi-equivariant for the action of . This requires a construction due to Schwartz [Sch92].
Theorem 1.11 (Bi-equivariant deformation retract).
There exists a deformation retract which is bi-equivariant for the action of , i.e. for every , and ,
In [Sch92], Schwartz constructs a deformation retract from to which is left-equivariant for the action of : for every , , and , . It occurs that this deformation retract is also right-equivariant by the action of : for every , , and every , we have .
Indeed, in order to construct the deformation retract, Schwartz considers smooth maps , with , evolving according to the following PDE:
where denotes the Schwarzian derivative. We recall that for every , and that satisfies the following cocycle relation:
in such a way that for every , we have:
Schwartz showed that the problem of finding satisfying the PDE (1.1) with a prescribed initial condition has a unique solution, and that moreover this solution approaches in the -topology as a unique function of the form , where is Möbius transform and we recall that denotes the exponential map. By the Invariance Relations (1.2), and the uniqueness of the solution of the Cauchy problem, it comes that for every and every , . In particular .
By conjugating this flow by the exponential map, Schwartz gets a deformation retract from to which is shown to be bi-equivariant for the action of . Recall that there is an identification which identifies with and with . Now the polar decomposition provides a deformation retract from to which is also bi-equivariant.
Recall that for a matrix , there is a unique pair of matrices such as , where and is a positive definite symmetric matrix with determinant . By the Spectral Theorem, such a matrix writes as , for a matrix , and a positive number (we use the notation for the diagonal matrix with entries , ). We have naturally a retraction sending to , defined by .
This retraction is clearly bi-equivariant by the action of , and thus passes to the quotient by : it gives a retraction which is bi-equivariant for the action of . From this, we deduce the desired bi-equivariant retraction from to .
Concatenating these two retractions, we get a bi-equivariant deformation retract from to . ∎
Deformation of the tangent bundle.
We can now use the deformation retract in order to get our continuous symmetric path of metrics.
Consider the arc length parametrization defined with . Denote by . Define the identity isotopy on in polar coordinates by
where and is the bump function defined in Paragraph 1.4.3.
Note that this identity isotopy is independent of the choice of a given arc length parametrization, and depends only on the metric . Indeed, another arc length parametrization writes as for some rotation . It is enough to note that by the equivariance property of , we have that and .
Note moreover that each preserves the circles .
The parametrization of by Euclidean arc length is geodesic for the pullback metric . Moreover, if we set for , we have for every .
The first part is a direct consequence of the definition of the function above. It remains to prove the symmetry property.
Consider the metric : any length of arc parametrization of reads as , where is some rotation. Now notice that:
where is the homotopy corresponding to . We conclude that the symmetry property holds. ∎
The isotopy described above depends continuously on . In particular it varies continuously with respect to in the smooth topology.
If the metric is standard around then is the identity, and the isotopy is trivial (i.e. is the identity for all ). In particular in this case.
1.4.5 Deformation of the normal bundle
The next step is to perform a twist isotopy on the hyperbolic metric in order to render the Euclidean normal vector field to perpendicular to with respect to our deformed hyperbolic metric.
Consider the unit inward normal vector field for the metric and let be such that
for all . Notice that is never equal to zero since is everywhere orthogonal to the vector field .
Let be the flow of the vector field . We define an identity isotopy on in polar coordinates by
where is as in the previous subsection, and .
Let be the pullback metric. Then the Euclidean normal vector field to is orthogonal to with respect to and the parametrization of by Euclidean arc length is a geodesic for .
The second claim follows because is the identity on for all .
Notice that on some neighborhood of . Hence the radial derivative of at a point is
The first claim follows because the right hand side is normal to with respect to . ∎
The isotopy depends only on the normal vector of and hence depends continuously on in the smooth topology.
If the metric is standard in a neighborhood of then one has so that is the trivial flow. Hence the isotopy just described leaves invariant in this case.
Symmetry of the procedure.
It remains to prove that the isotopy we described is symmetric. This is the content of the following lemma:
If we set for , , then we have that for every .
Denote by the objects associated to . First, note that preserves the circles centered at the origin. Hence, it is enough to prove that one has: .
But we see quite easily that the relation holds, first because , and then because by definition . Hence, it comes that : we deduce that conjugates the two flows. ∎
The -norm of any unit Euclidean tangent vector to is where is the length of with respect to . However the -norm of a Euclidean normal vector to varies from point to point. The purpose of this subsection is to deform to a metric such that both Euclidean normal and tangent vectors have the same -norm.
For this purpose let be the inward pointing vector field which has norm and is orthogonal to with respect to . Fix so that and define a smooth identity isotopy on in polar coordinates by
where is the push diffeomorphism defined in Paragraph 1.4.3.
The pullback metric is conformal with respect to the Euclidean metric on .
Since is the identity on one obtains that the Euclidean unit tangent vector to has -norm .
On the other hand the image of the Euclidean unit normal vector to at any point is exactly which has -norm equal to and is perpendicular to with respect to . This establishes the claim. ∎
The isotopy depends smoothly on the metric .
If is standard on a neighborhood of then is constant and equal to , hence the isotopy is the identity in this case.
Symmetry of the procedure.
It remains to prove that the procedure we described is symmetric.
If we set for , , then we have that for every .
Denote by the objects associated to . Here again, it is enough to prove that . By definition, we have , in such a way that : this enough to ensure the conjugacy formula. ∎
Collar Lemma and Fermi coordinates.
Consider the standard metric around (where is the length of with respect to ) and notice that coincides with on . Define the collar function as:
The Collar Lemma (see [Bus92]) asserts that the function that we will describe below is well defined on the round annulus formed by the points of the standard hyperbolic annulus with -distance to less than , where is the distance between and the geodesic . Note that this function varies continuously with the metric .
The function is defined as follows. The circle is mapped onto itself by the identity (here we use that the arc length parametrization of by is Euclidean: in the general case, we would have to use a more general differomorphism). Given , where , let be the -distance between and . Then is the point of which is at -distance along the -geodesic passing through perpendicular to . We just described the so-called Fermi coordinates [Bus92].
Note that a priori the Fermi coordinates depend on the choice of an arc length parametrization of by a rotation. Since is rotationally invariant, the definition of as is coherent.
Interpolation between the identity and the Fermi coordinates.
Let be the maximum of the first order derivatives of the coefficients of the metric on . The constants below can be chosen to be continuous functions of and .
Let be positive and such that the following properties hold.
the metrics and are bi-Lipschitz with Lipschitz constant less than or equal to on the annulus (use here that the two metrics coincide in ). In particular is defined on this annulus.
The distance between the two boundaries of is less than or equal to half of the corresponding distance between the boundaries of .
The distance between the two boundaries of is less than or equal to in such a way that is well defined on .
Notice that is the identity on and furthermore, because and coincide on , one has that the differential is the identity map at all points of . Furthermore is an isometry between restricted to and restricted to (see the first two items).
For some to be chosen later we define:
where is the bump function of Paragraph 1.4.3. Note that is in a neighborhood of the unit circle and is outside