Asymptotics of a class of Weil-Petersson geodesics and divergence of Weil-Petersson geodesics
We show that the strong asymptotic class of Weil-Petersson (WP) geodesic rays with narrow end invariant and bounded annular coefficients is determined by the forward ending laminations of the geodesic rays. This generalizes the Recurrent Ending Lamination Theorem of Brock, Masur and Minsky. As an application we provide a symbolic condition for divergence of WP geodesic rays in the moduli space.
2010 Mathematics Subject Classification:Primary 30F60, 32G15, Secondary 37D40
- 1 Introduction
- 2 Background
- 3 Combinatorial control
- 4 Variation of geodesics
- 5 Contraction property of WP geodesic segments
- 6 Strongly asymptotic rays
The Weil-Petersson (WP) metric is a Riemannian metric on the moduli space of Riemann surfaces. Over the last decade various aspects of the geometry and dynamics of the metric have been studied, however in comparison with the Teichmüller metric– the most well studied metric on the moduli space– much less is known about this metric. The WP metric is incomplete with sectional curvatures asymptotic to and in the completion. These features, in particular, prevent applying some of the standard techniques to study the global geometry and dynamics of the metric. For example, the Shadow Lemma for construction geodesic rays with specific behavior. WP geodesic rays are not necessarily visible [brockcommunication]. The Weil-Petersson geodesic flow is not uniformly hyperbolic. Also, curently Markov partitions are not available for coding of the WP geodesic flow.
Brock, Masur and Minsky [bmm1], in analogy with the vertical geodesic lamination of a Teichmüller geodesic, introduced a notion of ending lamination for Weil-Petersson geodesic rays. They conjectured that ending laminations, or a modification of them, can be used to parametrize the visual boundary of the WP metric and also the stable and unstable foliations of the WP geodesic flow. Furthermore, it is conjectured that end invariants and the associated subsurface coefficients provide a kind of symbolic coding for Weil-Petersson geodesics in the moduli space.
Further, Brock, Masur and Minsky [bmm1], [bmm2] explored several aspects of the mentioned conjectures. Significantly, they proved that the forward ending lamination determines the strong asymptotic class of recurrent WP geodesic rays to the thick part of the moduli space. Moreover, they showed that the bounded combinatorics of end invariants is equivalent to co-boundedness of the geodesic; the geodesic projects to a compact subset of the moduli space. These results have dynamical consequences, among which are the topological transitivity of the Weil-Petersson geodesic flow on the moduli space and unboundedness the topological entropy of the WP flow.
In [wpbehavior] we considered WP geodesics with narrow end invariants, end invariants with a certain constraint on subsurfaces with a big subsurface coefficient (see Definition 2.3), and constructed examples of closed WP geodesics in the thin parts of the moduli space as well as divergent WP geodesic rays with minimal filling ending laminations.
In this paper we show that the strong asymptotic class of a WP geodesic ray with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination.
(Narrow ending lamination theorem) The strong asymptotic class of a WP geodesic ray with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination.
The strong asymptotic class of a geodesic ray is the set of all the rays with as . The class of WP geodesic rays with narrow ending invariant and bounded annular coefficients contains geodesic rays which are not recurrent to any compact subset of the moduli space (divergent rays); see [wpbehavior, §8]. Heuristically these geodesic rays avoid all asymptotic flats in the WP metric and exhibit features of geodesics in manifolds with negative sectional curvatures which are bounded away from . This theorem is a generalization of the following result from [bmm1].
(Recurrent ending lamination theorem) The strong asymptotic class of a WP geodesic ray recurrent to a compact subset of the moduli space is determined by its forward ending lamination.
These theorems address the parametrization of the visual boundary of the WP metric and characterization of the stable and unstable foliations of the WP geodesic flow using laminations.
For the proof of Theorem 1.1 we use the control of the length-functions along WP geodesics developed in [wpbehavior] and ruled surfaces as in [bmm1]. The new ingredient here is the strict uniform contraction property of the nearest point projection to WP geodesic segments close to the thick part of a stratum which is not the product of lower complexity strata; see 5, in particular, Theorems 5.1 and 5.14. The contraction property is proved using some of Wolpert’s estimates on the WP metric, the WP Levi-Civita covariant derivatives and sectional curvatures in the thin part of the Teichmüller space and compactness arguments.
For WP geodesic rays with prescribed itinerary, as in Theorem 3.1, using the contraction property we can guarantee the existence of regions with a definite negative total curvature on ruled surfaces with one side on the ray with prescribed itinerary; see and the proof of Theorem 6.2.
A geodesic ray in a metric space is visible if for any other geodesic ray there is an infinite geodesic (strongly) asymptotic to in the forward time and (strongly) asymptotic to in the backward time. In a complete Riemannian manifold with negative sectional curvatures bounded away from every geodesic ray is visible. For the notion of visibility and some of its dynamical consequences see [ebergoedflownegcurv]. In the regions with a definite negative total curvature as above we are able to pick up enough negative curvature on the ruled surface so that using a variation of the Gauss-Bonnet formula asymptotic convergence to the ray with prescribed itinerary is guaranteed (Theorem 6.2). Using a similar technique, we overcome the difficulty caused by the fact that the sectional curvatures of the WP metric are not bounded away from in the thin part of the Teichmüller space, and we prove visibility of the class of geodesic rays with narrow end invariant and bounded annular coefficients in Theorem 6.5.
Finally, as an application of our ending lamination theorem we prove a symbolic condition in terms of subsurface coefficients for divergence of WP geodesic rays in the moduli space:
(Divergence condition) Given . Let be an narrow pair on a Riemann surface with bounded annular coefficients, and suppose that . Then a WP geodesic ray with end invariant is divergent in the moduli space .
Acknowledgement: I am so grateful to Yair Minsky for many invaluable discussions in the course of this work. I would also like to thank Scott Wolpert for several communications related to this work.
Let be two functions. Let and be two constants. We write if
holds for every .
2.1. Curve complexes and hierarchy paths
Let be a finite type, closed, orientable surface with genus and punctures or boundary components. Define the complexity of by . A subsurface of is an embedded, closed subsurface of with non-perpheral boundary curves.
The curve complex of the surface , denoted by , is a flag complex which serves to organize isotopy classes of simple closed curves on a surface. The complex is defined as follows: When , each vertex of the complex is the isotopy class of an essential simple closed curve, with an edge between each pair of isotopy classes with disjoint representatives on . In the same fashion there is a simplex corresponding to any set of pair-wise simple closed curves on . When , is or . The definition of the curve complex is the same, except that there is an edge between any pair of isotopy classes of curves with representatives with intersection number in the case of and in the case of .
When is an annulus with essential core curve the definition is slightly different. Let be the annular cover of to which lifts homeomorphically. There is a natural compactification of to a closed annulus which is obtained from the compactification of the Poincare disk (the universal cover of ) by the closed disk. A vertex of is associated to an arc connecting the two boundary components of modulo isotopies that fix the endpoints (isotopy classes of arcs relative to the boundary). There is an edge between two vertices which have representatives with disjoint interiors.
We equip the curve complex with a distance by making each simplex Euclidean with side lengths 1, and denote the distance by . One can easily verify that the curve complex of any annular subsurface is quasi-isometric to . Morever Masur and Minsky in their seminal work [mm1] proved that the curve complex of is hyperbolic with depending only on the topological type of .
A multi-curve on the surface is a collection of pair wise disjoint curves. Let and be two multi-curves. We say that and overlap and write , if there are curves and that intersect each other essentially (i.e. can not be realized as disjoint curves on ).
Laminations: Fix a complete finite area hyperbolic metric on . A geodesic lamination on is a closed subset of consisting of complete, simple geodesics. In particular, . Geodesic laminations provide a natural completion for the curve complex and Teichmüller space.
Each geodesic lamination can be equipped with a transverse measure [phtraintr]. The pair of a lamination and a transverse measure is called a measured geodesic lamination. We denote the space of measured geodesic laminations equipped with the weak topology by . acts on by rescaling measures, each equivalence class is called a projective measured lamination. We denote the quotient space by .
Recall that the curve complex of is Gromov hyperbolic. By a result of Klarreich [bdrycc] the Gromov boundary of the curve complex is identified with the ending laminations space . The space is the image of projective measured laminations with minimal filling support in under the measure forgetful map. Moreover, is equipped with the topology induced from the topology of via the measure forgetful map.
Pants decomposition and markings: A pants decomposition on is a maximal set of pair wise disjoint curves on . A (partial) marking is obtained from a pants decomposition by adding transversal curves to (some) all of the curve in the pants decomposition. We call the base of and denote it by . The set of all pants decompositions can be turned into a metric graph which is called the pants graph. For this purpose we put a length one edge between any two pants decompositions which differ by an elementary move. Similarly the markings can be turned into a metric graph called the marking graph. For more detail see [mm2].
Subsurface coefficient: An essential subsurface is a compact, connected subsurface of whose boundary consists of essential curves in or boundary curves of which is not a holed sphere. In this paper we do not distinguish between a subsurface and its isotopy class.
Let be an essential subsurface. We define the subsurface projection map
where is the power set of as follows: Equip with a complete hyperbolic metric and realize all curves and laminations geodesically. Let . Suppose that is a non-annular subsurface. If , then define . Otherwise, consider the set of arcs with end points on or at cusps in the intersection locus . Identify all of the curves and arcs that are isotopic. Through these isotopies the end points of arcs are allowed to move within . The projection is the union of the boundary curves of a regular neighborhood of where is an arc we obtained above, and all the closed curves we obtained above.
For an annulus the subsurface projection is the set of component arcs of the lift of to (the compactification of the annular cover of to which lifts homeomorphically) that connect the two boundaries of . For more detail see [mm2, §2].
The projection of a multi-curve to a subsurface is the union of the projections of all . Let be a partial marking. If is an annular subsurface with core curve , then is the set of transversal curves of . Otherwise, .
The subsurface coefficient of two laminations or (partial) markings and is defined by
Note that the subsurface coefficients are an analogue of continued fraction expansions of real numbers. We use subsurface coefficients to study geodesics on moduli spaces, similar to the role of continued fraction expansion in the study of geodesics on the modular surface.
We denote by the diameter of viewed as a subset of . The following lemma is a straightforward consequence of [mm2, Lemma 2.3].
Let be a (partial) marking on a surface . For any essential subsurface we have
The hierarchy paths introduced by Masur and Minsky [mm2] comprise a transitive family of quasi-geodesics in the pants graph of a surface with quantifiers depending only the topological type of the surface. These quasi-geodesics are constructed from hierarchies of geodesics in subsurfaces of the surface. The main feature of hierarchy paths is that their properties are encoded in their end points and the associated subsurface coefficients. For a list of the properties of hierarchy paths see [bmm2], [wpbehavior].
Let be a pair of (partial) markings or laminations. Let , where , be a hierarchy path with and . An important property of the hierarchy path is the following: There are subsurfaces called component domains; corresponding to each component domain there is a connected subinterval such that for all . Moreover there is a constant depending only on the topological type of the surface, so that any subsurface with is a component domain.
Using the machinery of hierarchies Masur and Minsky [mm2] established the following quasi-distance formula:
Given , there exist constants and such that for any we have
Here the cut off function is defined by We call the threshold constant and and the corresponding multiplicative and additive constants.
In the following we define two constraints on subsurface coefficients which would be used in this paper.
(Narrow pair) Let . An narrow pair of (partial) markings or laminations is a pair such that for an essential subsurface if
then is large, i.e. each connected component of is an annulus or a three holed sphere. In [wpbehavior] we proved that any hierarchy path between a narrow pair is stable in the pants graph.
(Bounded combinatorics) Given . Let be a hierarchy path. Let . We say that has non-annular bounded combinatorics over if
for every essential non-annular subsurface . Moreover,
for every .
2.2. Weil-Petersson metric
Let be a surface with only punctures (no boundary curves). A point in the Teichmüller space of the surface , denote by , is a complete finite area hyperbolic surface equipped with a diffeomorphism . The diffeomorphism is a marking of . Two marked surfaces and define the same point in if and only if is isotopic to an isometry. The moduli space of , denoted by , is the quotient of by the action of the mapping class group of , denoted by , on by precomposition (remarking).
The thick part of the Teichmüller space consists of all with injectivity radius . The thin part consists of all with . Suppose that is small enough that by the Collar Lemma (see [buser, §4.1]) there is no pair of intersecting closed geodesics of length less than or equal to on a complete hyperbolic surface. Given a multi-curve we define the open regions in Teichmüller space
For a comprehensive introduction to the Weil-Petersson metric and its properties we refer the reader to [wol],[wolb]. Here we only recall some of the properties of the metric which are important for us. The Weil-Petersson (WP) metric is a Riemannian metric with negative sectional curvatures on the Teichmüller space. The metric is incomplete due to possibility of pinching curves along paths with finite WP length. However there is a unique WP geodesic between any two points in the Teichmüller space so the metric is geodesically convex. The WP sectional curvatures are negative and asymptotic to both and asymptotic to completion. The completion is a metric. The metric is invariant under the action of and descends under the natural orbifold cover to a metric on .
The Weil-Petersson completion of the Teichmüller space is the disjoint union of strata denoted by where is a multi-curve. consists of nodal Riemann surfaces at . Equivalently each point in is a marked complete hyperbolic surface with a pair of cusps for each curve in . The WP completion of the Teichmüller space descends to the Deligne-Mumford compactification of the moduli space via the action of the mapping class group. is the union of finitely many strata. Each stratum is the quotient of the strata of which are identified by the natural extension of the action of to the completion.
The WP metric has the following non-refraction property: Given points , the interior of the unique geodesic connecting and is in the smallest stratum (with respect to inclusion of strata) that contains and . The following strengthened version of Wolpert’s Geodesic Limit Theorem (see [wols] and [bmm2]) proved in of [wpbehavior] provides a limiting picture for a sequence of bounded length WP geodesic segments in the Teichmüller space. We need this result for compactness arguments in .
(Geodesic limits) Given , let be a sequence of WP geodesic segments parametrized by arc-length. After possibly passing to a subsequence, there are a partition of , possibly empty multi-curves , a multi-curve for , and a piece-wise geodesic
with the following properties
Given a multi-curve denote by the subgroup of generated by positive Dehn twists about the curves in . There are elements of the mapping class group for each so that either or is an unbounded sequence, and for and any so that
For any , as . For each and each , let . Then for any , , as .
2.3. End invariants
There exists a constant (Bers constant) depending only on the topological type of the surface with the property that any complete finite area hyperbolic metric on has a pants decomposition (Bers pants decomposition) such that the length of every curve in is at most ; see [buser, §5]. A Bers curve is a curve in a Bers pants decomposition. A Bers marking is a (partial) marking obtained from a Bers pants decomposition by adding transversal curves with representatives of minimal length. Given a point , suppose that . A Bers pants decomposition of , denoted by , is the union of Bers pants decompositions of the connected components of and the multi-curve . A Bers marking of , denoted by , is obtained from with no transversal for curves in .
By Brock’s Quasi-Isometry Theorem [br] the coarse map
which assigns to a Bers pants decomposition of is a quasi-isometry with constants and depending only on the topological type of .
(Ending measured lamination) The weak limit in of any infinite sequence of weighted, distinct Bers curves along a WP geodesic ray is an ending measured lamination of .
For any , the length-function assigns to a point the length of the geodesic representative of on the hyperbolic surface . This notion of length-function has a natural extension to the space of measured laminations . Let . We denote the value of the length-function at a point by by .
The convexity of length-functions along WP geodesics proved by Wolpert (see e.g. [wol, §3]) asserts that
Let be a WP geodesic. For any , is a convex function. Similarly, for any , is a convex function.
In [bmm2] the following notion of ending lamination for WP geodesic rays is introduced. Its existence relies on the convexity of length-functions along WP geodesics and properties of spaces. Let be a WP geodesic ray.
(Ending Lamination) The union of pinching curves along a WP geodesic ray and the geodesic laminations arising as supports of all ending measured laminations of is the ending lamination of . Where a pinching curve of is a curve such that as .
(End invariant of Weil-Petersson geodesics) To each open end of a geodesic (we assume that ) we associate an end invariant which is a partial marking or a lamination. If the forward trajectory can be extended to such that then the forward end invariant is any Bers marking (there are finitely many of them). Otherwise, is the ending lamination of the forward trajectory ray which was defined above. We define the backward end invariant similarly by considering the backward trajectory . We call the pair the end invariant of .
We recall two important properties of the ending measured laminations proved in [bmm1, §2].
(Decreasing of length along WP geodesic rays) Let be any ending measured lamination of a WP geodesic ray , then is a decreasing function.
Let be a convergent sequence of rays in the WP visual sphere at . Then if is any sequence of ending measured laminations or weighted pinching curves for , any representative of the limit of the projective classes in has bounded length along the ray .
Let be a measurable geodesic lamination. Suppose that there is a collection of pairwise disjoint subsurfaces , , with so that any simple closed curve in is isotopic to a boundary curve of one of the subsurfaces , and moreover that the restriction of to is minimal and fills . For each let be a measured lamination supported on . Let be a sequence of curves so that the projective classes converge to in as . For each , let be a pants decomposition that contains . Let be the maximally nodal hyperbolic surface at . Let be the WP geodesic segment connecting a base point in the interior of the Teichmüller space to . Denote the parametrization of by arc-length by . In [wpbehavior, §8] we proved
(Infinite ray) After possibly passing to a subsequence the geodesic segments converge to an infinite ray in the visual sphere of the WP metric at . Moreover, the length of each measured lamination , , and each curve is bounded along .
Let be a narrow pair. The narrow condition implies that there is at most one subsurface with so that the restriction of to is minimal and fills . Suppose that such a component exists. Let be a hierarchy between and . There is an so that for all we have
Note that if , then this statement vacuously holds. For each , let and be a curve in . After possibly passing to a subsequence the projective classes of converge to the projective class of a measured geodesic lamination supported on . Let be a point with a Bers marking . Let be the maximally nodal hyperbolic surface at . As before, let be the limit of the geodesic segments after possibly passing to a subsequence. In [wpbehavior, §8] we proved:
The forward ending lamination of contains .
3. Combinatorial control
Let be a WP geodesic with narrow end invariant . Let be a hierarchy path between and . In [wpbehavior, §5] we proved that a hierarchy path with narrow end points is stable. Where is the quantifier function of the stability which depends only on . Thus and , fellow-travel each other where depends only on and the topological type of . Moreover, since both and are quasi-geodesics with quantifiers depending only on the topological type of the surface , there is a coarse parameter map from to such that
where the constants and depend only on ; see [wpbehavior, §5.3].
The following theorem from [wpbehavior, §8] provides us a WP geodesic ray with a prescribed itinerary in the Teichmüller space.
(Infinite ray with prescribed itinerary) Given , there are constants and with the following properties.
Let be an narrow pair. Let be a hierarchy path between and . Let be the infinite WP geodesic ray as in Lemma 2.13. Suppose that a large component domain of has bounded combinatorics over an interval with . Let and . Then for every we have
for every , and
Moreover, if and are subsurfaces as above, implies that .
3.1. Bounding annular coefficients
Let be a WP geodesic ray with prescribed itinerary where the end invariant is narrow and has bounded annular coefficients. To goal of this section is to prove Lemma 3.6. Where show that over any long enough subinterval of , there is a subinterval of definite length over which is in some region of the Teichmüller space ( regions were defined in ). This combinatorial control will be used in 5.
First we recall two properties of hierarchy paths which will be used in this section. For an extended list of properties of hierarchy paths see [wpbehavior, §2], [bmm2, §2].
There exist positive constants and depending only on the topological type of with the following properties. Let be a hierarchy path between partial markings or laminations and . Let be a component domain of and let . Then
for any , and for any .
(No backtracking) Let with . Then for any subsurface ,
Given an increasing function , there is an , depending only on and the topological type of with the following property. Let be a hierarchy path. Suppose that a subinterval has the property that for any subsurface and any , if has non-annular bounded combinatorics over a subinterval , then . Then we have that .
Let be the complexity of the surface . For each define the constant
Note that for , the set on the right-hand side is empty and we define .
Let . For any essential subsurface with we have
The proof of the claim is by contradiction. Suppose that the claim does not hold. Then there is a non-annular subsurface with such that
Suppose that (3.3) holds. Then .
Let . To get a contradiction, suppose that the claim does not hold. Then we have that . This implies that either or .
By Claim 3.5 we have that . Define the interval
We proceed to show that
If , then by the triangle inequality and Lemma 2.2,
Now the above two inequalities and the inequality (3.3) together imply that the inequality
holds, from which the bound (3.4) follows.
If , then as above we can get the bound
which gives us the bound (3.4).
The above two inequalities and (3.3) combined with the triangle imply that
But this contradicts the assumption of the lemma. Claim 3.4 follows from this contradiction.
By Claim 3.4, for each , we have
moreover from the setup of the constants in (3.2), it is clear that
so we have the bound
To simplify the notation we define the function
By (3.5), . Also we have that . Then we may inductively show that
for each , where denotes the th composition of with itself.
Since is an increasing function and , we have that
Then using the bound (3.6) we see that
So by the setup of the constants in (3.2) we have
Moreover, is a quasi-geodesic where and depend only on the topological type of , so we obtain the upper bound for . Note that only depends on and the topological type of . The proof of the lemma is complete. ∎
Let be the constant from Theorem 3.1.
Given positive constants and , there are constants and with the following properties. Let be an narrow pair with bounded annular coefficients. Let be a hierarchy path between and . Let be a WP geodesic ray with prescribed itinerary and end invariant . Then for any subinterval with , there are a subinterval and a large component domain of such that for and ( is the parameter correspondence map) we have for every . Moreover, .
Fix a threshold constant for the distance formula (2.2) and let be the corresponding constants. Note that the hierarchy path is a quasi-geodesic where depend only on the topological type of the surface. Let and . Let be the constants for from (3.1). Let be the constant from Theorem 3.2(2). Let be the constant from Theorem 3.1. Define the function
and has non-annular bounded combinatorics over . Since is narrow and , the subsurface is a large subsurface. Thus for any non-annular subsurface , either or holds. If , then since , we have . If , then by the non-annular bounded combinatorics we have . Then by the distance formula (2.2) and the fact that has bounded combinatorics over , we have
As we saw above , so
Moreover, is a quasi-geodesic, so
Therefore, has bounded combinatorics over . Set . Then the lemma follows from applying Theorem 3.1 to the interval . ∎
4. Variation of geodesics
Let be a geodesically convex, negatively curved, Riemannian manifold; for example the Teichmüller space equipped with the WP metric. Let be a geodesic segment, and let be the nearest point projection from to . In the following proposition we collect some important facts about the map which we need:
Suppose that is a point in so that is in the interior of or is an end point or which is the nearest point to on a slightly longer geodesic segment containing . Let be a geodesic segment connecting and . We have
The projection map is continuous at .
If is not on , then is smooth at .
The distance function is smooth.
Part (1) is [mfldnc, Lemma 3.2]. Part (2) follows from [chebcomparison, Proposition 1.7]. Part (3) follows from the lemma on the first page of [regdist]. Part (4) is [regdist, Theorem 1]. ∎
Let and be two geodesic segments parametrized by arclength. Let be a point whose nearest point projection on is in the interior of . Note that is a smooth path, moreover by Proposition 4.1(4) the end point of varies smoothly. Therefore is a smooth family of geodesic segments with respect to . Moreover, since is a negatively curved manifold, any two of the geodesic segments fellow travel all the way. Finally, let be the reparametrization of which maps to . Then is smooth.