In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson:
Theorem 1 (Burns-Masur-Wilkinson) Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously. Denote by the metric completion of and the boundary of .Suppose that:
- (I) the universal cover of is geodesically convex, i.e., for every , there exists an unique geodesic segment in connecting and .
- (II) the metric completion of is compact.
- (III) the boundary is volumetrically cusplike, i.e., for some constants and , the volume of a -neighborhood of the boundary satisfies
for every .
- (IV) has polynomially controlled curvature, i.e., there are constants and such that the curvature tensor of and its first two derivatives satisfy the following polynomial bound
for every .
- (V) has polynomially controlled injectivity radius, i.e., there are constants and such that
for every (where denotes the injectivity radius at ).
- (VI) The first derivative of the geodesic flow is polynomially controlled, i.e., there are constants and such that, for every infinite geodesic on and every :
Then, the Liouville (volume) measure of is finite, the geodesic flow on the unit cotangent bundle of is defined at -almost every point for all time , and the geodesic flow is non-uniformly hyperbolic (in the sense of Pesin’s theory) and ergodic.
Actually, the geodesic flow is Bernoulli and, furthermore, its metric entropy is positive, finite and is given by Pesin’s entropy formula (i.e., it is equal to the sum of positive Lyapunov exponents of counted with multiplicities).
However, since the second post of this series was dedicated to the discussion of items (I), (II) and (III) above for the Weil-Petersson (WP) metric, we think it is natural that this third post provides a discussion of items (IV), (V) and (VI) for the Weil-Petersson metric (thus completing the proof of Burns-Masur-Wilkinson theorem of ergodicity of the Weil-Petersson geodesic flow modulo the proof of their ergodicity criterion).
For this reason, we will continue the discussion of the geometry of the Weil-Petersson metric in this post while leaving the proof of Burns-Masur-Wilkinson ergodicity criterion for the next two posts of this series.
The organization of today’s post is very simple: it is divided in three sections where the items (IV), (V) and (VI) for the Weil-Petersson metric are discussed.
1. The curvatures of the Weil-Petersson metric
The item (IV) of Burns-Masur-Wilkinson ergodicity criterion (Theorem 1) asks for polynomial bounds in the sectional curvatures and their first two derivatives.
In the context of the Weil-Petersson (WP) metric, the desired polynomial bounds on the sectional curvatures follow from the work of Wolpert.
1.1. Wolpert’s formulas for the curvatures of the WP metric
This subsection gives a compte rendu of some estimates of Wolpert for the behavior of the WP metric near the boundary of the Teichmüller space .
Before stating Wolpert’s formulas, we need an adapted system of coordinates (called combined length basis in the literature) near the strata , , of , where is the curve complex of (introduced in the previous post).
Denote by the set of pairs (“basis”) where is a simplex of the curve complex and is a collection of simple closed curves such that each is disjoint from all . Here, we allow that two curves intersect (i.e., one might have ) and also the case is not excluded.
Following the nomenclature introduced by Wolpert, we say that is a combined length basis at a point whenever the set of tangent vectors
is a basis of , where is the length parameter in the Fenchel-Nielsen coordinates and .
Remark 1 The length parameters and their square-roots are natural for the study of the WP metric: for instance, Wolpert showed that these functions are convex along WP geodesics (see, e.g., these papers of Wolpert and this paper of Wolf).
The name combined length basis comes from the fact that we think of as a combination of a collection of short curves (indicating the boundary stratum that one is close to), and a collection of relative curves to allowing to complete the set into a basis of the tangent space to in which one can write nice formulas for the WP metric.
This notion can be “extended” to a stratum of as follows. We say is a relative basis at a point whenever and the length parameters is a local system of coordinates for near .
Remark 2 The stratum is (isomorphic to) a product of the Teichmüller spaces of the pieces of . In particular, carries a “WP metric”, namely, the product of the WP metrics on the Teichmüller spaces of the pieces of . In this setting, is a relative basis at if and only if is a basis of .
Remark 3 Contrary to the Fenchel-Nielsen coordinates, the length parameters associated to a relative basis might not be a global system of coordinates for . Indeed, this is so because we allow the curves in to intersect non-trivially: geometrically, this means that there are points in where the geodesic representatives of such curves meet orthogonally, and, at such points , the system of coordinates induced by meet a singularity.
The relevance of the concept of combined length basis to the study of the WP metric is explained by the following theorem of Wolpert:
Theorem 2 (Wolpert) For any point , , there exists a relative length basis . Furthermore, the WP metric can be written as
where the implied comparison constant is uniform in a neighborhood of . In particular, there exists a neighborhood of such that is a combined length basis at any .
The statement above is just the beginning of a series of formulas of Wolpert for the WP metric and its sectional curvatures written in terms of the local system of coordinates induced by a combined length basis .
In order to write down the next list of formulas of Wolpert, we need the following notations. Given an arbitrary collection of simple closed curves on , we define
where . Also, given a constant and a basis , we will consider the following (Bers) region of Teichmüller space:
Wolpert provides several estimates for the WP metric and its sectional curvatures in terms of the basis , and , , which are uniform on the regions .
- where is the Kronecker delta.
- and, furthermore, extends continuosly to the boundary stratum .
- the distance from to the boundary stratum is
- for any vector ,
- extends continuously to the boundary stratum
- the sectional curvature of the complex line (real two-plane) is
- for any quadruple , distinct from a curvature-preserving permutation of , one has
and, moreover, each of the form or introduces a multiplicative factor in the estimate above.
These estimates of Wolpert gives a very good understanding of the geometry of the WP metric in terms of combined length basis. For instance, one infers from the last two items above that the sectional curvatures of the WP metric along the complex lines converge to with speed as one approaches the boundary stratum , while the sectional curvatures of the WP metric associated to quadruples of the form with and converge to with speed at least.
In particular, these formulas of Wolpert allow to show “1/3 of item (IV)” for the WP metric, that is,
for all .
Remark 4 Observe that the formulas of Wolpert provide asymmetric information on the sectional curvatures of the WP metric: indeed, while we have precise estimates on how these sectional curvarutures can approach , the same is not true for the sectional curvatures approaching zero (where one disposes of lower bounds but no upper bounds for the speed of convergence).
Remark 5 From the discussion above, we see that there are sectional curvatures of the WP metric on approaching zero whenever contains two distinct curves. In other words, the WP metric has sectional curvatures approaching zero whenever the genus and the number of punctures of satisfy , i.e., except in the cases of once-punctured torii and four-times puncture spheres . This qualitative difference on the geometry of the WP metric on in the cases and (i.e., or ) will be important in the last post of this series when we will discuss the rates of mixing of the WP geodesic flow.
Remark 6 As Wolpert points out in this paper here, these estimates permit to think of the WP metric on the moduli space in a -neighborhood of the cusp at infinity as a -pertubation of the metric of the surface of revolution of the profile modulo multiplicative factors of the form .
Now, we will investigate the remaining “2/3 of item (IV)” for the WP metric, i.e., polynomial bounds for the first two derivatives and of the curvature operator of the WP metric.
1.2. Bounds for the first two derivatives of WP metric
As it was recently pointed out to us by Wolpert (in a private communication), it is possible to deduce very good bounds for the derivatives of the WP metric (and its curvature tensor) by refining the formulas for the WP metric in some of his works.
Nevertheless, by the time the article of Burns, Masur and Wilkinson was written, it was not clear at all that the delicate calculations of Wolpert for the WP metric could be extended to provide useful information about the derivatives of this metric.
For this reason, Burns, Masur and Wilkinson decided to implement the following alternative strategy.
At first sight, our task reminds the setting of Cauchy’s inequality in Complex Analysis where one estimates the derivatives of a holomorphic function in terms of given bounds for the -norm of this function via the Cauchy integral formula. In fact, our current goal is to estimate the first two derivatives of a “function” (actually, the curvature tensor of the WP metric) defined on the complex-analytic manifold knowing that this “function” already has nice bounds (cf. the previous subsection).
However, one can not apply the argument described in the previous paragraph directly to the curvature tensor of the WP metric because this metric is only a real-analytic (but not a complex-analytic/holomorphic) object on the complex-analytic manifold .
Fortunately, Burns, Masur and Wilkinson observed that this idea of using the Cauchy inequalities could still work after one adds some results of McMullen into the picture. In a nutshell, McMullen showed that the WP metric is closely related to a holomorphic object: very roughly speaking, using the so-called Bers simultaneous uniformization theorem, one can think of the Teichmüller space as a totally real submanifold of the so-called quasi-Fuchsian locus , and, in this setting, the Weil-Petersson symplectic -form is the restriction to of the differential of a holomorphic -form globally defined on the quasi-Fuchsian locus . In particular, it is possible to use Cauchy’s inequalities to the holomorphic object to get some estimates for the first two derivatives of the WP metric.
Remark 7 A caricature of the previous paragraph is the following. We want to estimate the first two derivatives of a real-analytic function (“WP metric”) knowing some bounds for the values of . In principle, we can not do this by simply applying Cauchy’s estimates to , but in our context we know (“by the results of McMullen”) that the natural embedding of as a totally real submanifold of allows to think of as the restriction of a holomorphic function and, thus, we can apply Cauchy inequalities to to get some estimates for .
In what follows, we will explain the “Cauchy inequality” idea of Burns, Masur and Wilkinson in two steps. Firstly, we will describe the embedding of into the quasi-Fuchsian locus and the holomorphic -form of McMullen whose differential restricts to the WP symplectic -form on . After that, we will show how the Cauchy inequalities can be used to give the remaining “2/3 of item (IV)” for the WP metric.
1.2.1. Quasi-Fuchsian locus and McMullen’s -forms
Given a hyperbolic Riemann surface , , the quasi-Fuchsian locus is defined as
where is the conjugate Riemann surface of , i.e., is the quotient of the lower-half plane by . The Fuchsian locus is the image of under the anti-diagonal embedding
Geometrically, we can think of elements as follows. Recall that and are related to and via (extremal) quasiconformal mappings determined by the solutions of Beltrami equations associated to -invariant Beltrami differentials (coefficients) and on and . Now, we observe that and live naturally on the Riemann sphere . Since the real axis/circle at infinity/equator has zero Lebesgue measure, we see that and induce a Beltrami differential on . By solving the corresponding Beltrami equation, we obtain a quasiconformal map on and, by conjugating, we obtain a quasi-Fuchsian subgroup
i.e., a Kleinian subgroup whose domain of discontinuity consists of two connected components and such that and .
The following picture summarizes the discussion of the previous paragraph:
Remark 8 The Jordan curve given by the image of the equator under the quasiconformal map is “wild” in general, e.g., it has Hausdorff dimension (as the picture above tries to represent). In fact, this happens because a typical quasiconformal map is merely a Hölder continuous, and, hence, it might send “nice” curves (such as the equator) into curves with “intricate geometries” (see, e.g., the three external links of the Wikipedia article on quasi-Fuchsian groups).
The data of the quasi-Fuchsian subgroup attached to permits to assign (marked) projective structures to and . More precisely, by writing and with and , we are equipping and with projective structures, that is, atlases of charts to whose changes of coordinates are Möebius transformations (i.e., elements of ). Furthermore, by recalling that and come with markings and (because they are points in Teichmüller spaces), we see that the projective structures above are marked.
In summary, we have a natural quasi-Fuchsian uniformization map
assigning to the marked projective structures
Here, is the “Teichmüller space of projective structures” on , i.e., the space of “Teichmüller” equivalence classes of marked projective structures where two marked projective structures and are “Teichmüller” equivalent whenever there is a projective isomorphism homotopic to .
Remark 9 The procedure (due to Bers) of attaching a quasi-Fuchsian subgroup to a pair of hyperbolic surfaces and is called Bers simultaneous uniformization because the knowledge of allows to equip at the same time and with natural projective structures.
Note that is a section of the natural projection
obtained by sending each pair of (marked) projective structures , , , to the unique pair of (marked) compatible conformal structures , , .
We will now describe how the (affine) structure of the fibers of the projection and the section can be used to construct McMullen’s primitives/potentials of the Weil-Petersson symplectic form .
Given two projective structures in the same of the projection , one can measure how far apart from each other are and using the so-called Schwarzian derivative.
More precisely, the fact that and induce the same conformal structure means that the charts of atlases associated to them can be thought as some families of maps and from (small) open subsets to the Riemann sphere , and we can measure the “difference” by computing how “far” from a Möebius transformation (element of ) is .
Here, given a point , one observes that there exists an unique Möebius transformation such that and coincide at up to second order (i.e., and have the same value and the same first and second derivatives at ). Hence, it is natural to measure how far from a Möebius transformation is by understanding the difference between the third derivatives of and at , i.e., .
Actually, this is almost the definition of the Schwarzian derivative: since the derivatives of and map to , in order to recover an object from to itself, it is a better idea to “correct” with , i.e., we define the Schwarzian derivative of and at as
Here, the factor shows up for historical reasons (that is, this factor makes coincide with the classical definition of Schwarzian derivative in the literature).
By definition, the Schwarzian derivative is a field of quadratic forms on (since its definition involves taking third order derivatives). In other terms, is a quadratic differential on , that is, the “difference” between two projective structures in the same fiber of the projection is given by a quadratic differential . In particular, the fibers are affine spaces modeled by the space of quadratic differentials on .
Remark 10 The reader will find more explanations about the Schwarzian derivative in Section 6.3 of Hubbard’s book.
Remark 11 The idea of “measuring” the distance between projective structures (inducing the same conformal structure) by computing how far they are from Möebius transformations via the Schwarzian derivative is close in some sense to the idea of measuring the distance between two points in Teichmüller space by computing the eccentricities of quasiconformal maps between these points.
Using this affine structure on and the fact that is the cotangent space of at , we see that, for each , the map
defines a (holomorphic) -form on . Note that, by letting vary and by fixing , we have a map given by
Since (so that ) and , we can think of as a (holomorphic) -form on .
For later use, let us notice that the -form is bounded with respect to the Teichmüller metric on . Indeed, this is a consequence of Nehari’s bound stating that if is a round disc (i.e., the image of the unit disc under a Möebius transformation) equipped with its hyperbolic metric and is an injective complex-analytic map, then
In this setting, McMullen constructed primitives/potentials for the WP symplectic form as follows. The Teichmüller space sits in the quasi-Fuchsian locus as the Fuchsian locus where is the anti-diagonal embedding
By pulling back the -form under , we obtain a bounded -form
Remark 12 This form is closely related to a classical object in Teichmüller theory called Bers embedding: in our notation, the Bers embedding is
McMullen showed that the bounded -forms are primitives/potentials of the WP symplectic -form , i.e.,
See also Section 7.7 of Hubbard’s book for a nice exposition of this theorem of McMullen. Equivalently, the restriction of the holomorphic -form to the Fuchsian locus (a totally real sublocus of ) permits to construct (Teichmüller bounded) primitives for the WP symplectic form on .
At this point, we are ready to implement the “Cauchy estimate” idea of Burns-Masur-Wilkinson to deduce bounds for the first two derivatives of the curvature operator of the WP metric.
1.2.2. “Cauchy estimate” of after Burns-Masur-Wilkinson
Following Burns-Masur-Wilkinson, we will need the following local coordinates in :
of the Euclidean unit polydisc (where ) sending to and satisfying
where is the Teichmüller norm and is the Euclidean norm on .
This result is proven in this paper of McMullen here.
Also, since the statement of Proposition 4 involves the Teichmüller norm and we are interested in the Weil-Petersson norm , the following comparison (from Lemma 5.4 of Burns-Masur-Wilkinson paper) between and will be helpful:
where is the systole of (i.e., the length of the shortest closed simple hyperbolic geodesics on ). In particular, for any and any tangent vector , one has
Proof: Given , let us write with is “normalized” to contain the element where .
Fix a Dirichlet fundamental domain of the action of centered at the point .
By the collaring theorem stating that a closed simple hyperbolic geodesic of length has a collar [tubular neighborhood] of radius isometrically embedded in and two of these collars and are disjoint whenever and are disjoint (see, e.g., Theorem 3.8.3 in Hubbard’s book), we have that the union of isometric copies of contains a ball of fixed (universal) radius around any point .
By combining the Cauchy integral formula with the fact stated in the previous paragraph, we see that
Since the hyperbolic metric is bounded away from on , we can use the -norm estimate on above to deduce that
for some constant . This completes the proof of the lemma.
Remark 13 The factor in the previous lemma can be replaced by via a refinement of the argument above. However, we will not prove this here because this refined estimate is not needed for the proof of the main results of Burns-Masur-Wilkinson.
Using the local coordinates from Proposition 4 (and the comparison between Teichmüller and Weil-Petersson norms in the previous lemma), we are ready to use Cauchy’s inequalities to estimate “‘s” of the WP metric. More concretely, denoting by “centered at some ” in Proposition 4, let , and consider the vector fields
on . In setting, we denote by the “‘s” of the WP metric in the local coordinate and by the inverse of the matrix .
Proposition 6 There exists an universal constant such that, for any , the pullback of the WP metric local coordinate “centered at ” in Proposition 4 verifies the following estimates:
for all , and , .
Proof: The first inequality
we deduce that
For the proof of second inequality (estimates of the -derivatives of ‘s), we begin by “rephrasing” the construction of McMullen’s -form in terms of the local coordinate introduced in Proposition 4.
The composition of the local coordinate with the anti-diagonal embedding of the Teichmüller space in the quasi-Fuchsian locus can be rewritten as
where is the anti-diagonal embedding
and the local coordinate given by
In this setting, the pullback by of the holomorphic -form gives a holomorphic -form on . Moreover, since the Euclidean metric on is comparable to the pullback by of the Teichmüller metric (cf. Proposition 4), is bounded in Teichmüller metric and where , we see that
where and is a holomorphic bounded (in the Euclidean norm) -form on .
Let us write in complex coordinates , where are bounded holomorphic functions. Hence,
and, a fortiori,
Since is the Kähler form of the metric , we see that the coefficients of are linear combinations of the -pullbacks of and . Because are (universally) bounded holomorphic functions, we can use Cauchy’s inequalities to see that the derivatives of are (universally) bounded at any with . It follows from the boundedness of the (non-holomorphic) anti-diagonal embedding that the -derivatives of ‘s satisfy the desired bound.
Proof: Fix and consider the local coordinate provided by Proposition 4. Since and are uniformly bounded, our task is reduced to estimate the first two derivatives of the curvature tensor of the metric at the origin .
Recall that the Christoffel symbols of are
Therefore, we see that the coefficients of the -derivative is a polynomial function of and the first partial derivatives whose “degree” in the “variables” is (because of the formula ).
By Proposition 6, each has order and the first partial derivatives of at are bounded by a constant depending only on . It follows that
This completes the proof.
Remark 14 The estimates for the derivatives of the curvature tensor appearing in the proof of Theorem 7 are not sharp with respect to the exponent . For instance, the WP metric on the moduli space of once-punctured torii has curvature where is the WP distance between and the boundary , so that one expects tha the -derivatives of the curvature behave like (i.e., the exponent above should be ).In a very recent private communication, Wolpert indicated that it is possible to derive the sharp estimates of the form
for the derivatives of the curvature tensor of the WP metric from his works.
2. Injectivity radius of the Weil-Petersson metric
In this short section, we will verify item (V) of Burns-Masur-Wilkinson ergodicity criterion (Theorem 1) for the WP metric, i.e.,
The proof of this result also relies on the work of Wolpert. More precisely, Wolpert showed in this paper here that there exists a constant such that, for any and with ,
where is the Abelian subgroup of the “level ” mapping class group generated by the Dehn twists about the curves .
This reduces the proof of Theorem 8 to the following lemma:
Lemma 9 There exists an universal constant with the following property. For each , there exists such that, for any with
for some non-trivial , one can find so that and for some .
Proof: We begin the proof of the lemma by recalling that the mapping class group acts on in a properly discontinuous way with no fixed points. Therefore, for each , there exists such that if for some non-trivial (i.e., some non-trivial element of the mapping class group has an “almost fixed point”), then (i.e., the “almost fixed point” is close to the boundary of ).
Let us show now that in the setting of the previous paragraph, for some .
In this direction, let be the product of and the maximal orders of all finite order elements of the mapping class groups of “lower complexity” surfaces. By contradiction, let us assume that there exist infinite sequences , , , such that for some and
but for all , .
Passing to a subsequence (and applying appropriate elements of ), we can assume that the sequence converges to some noded Riemann surface . Because as , we see that ,for each ,
It follows that, for all sufficiently large, sends any curve to another curve . Therefore, for each sufficiently large, there exists
such that fixes each (i.e., is a reducible element of the mapping class group). By the Nielsen-Thruston classification of elements of the mapping class groups, the restrictions of to each piece of are given by compositions of Dehn twists about the boundary curves with either a pseudo-Anosov or a periodic (finite order) element (in a surface of “lower complexity” than ).
It follows that we have only two possibilities for : either the restriction of to all pieces of are compositions of Dehn twists about certain curves in and finite order elements, or the restriction of to some piece of is the composition of Dehn twists about certain curves in and a pseudo-Anosov element.
In the first scenario, by the definition of , we can replace by an adequate power with to “kill” the finite order elements and “keep” the Dehn twists. In other terms, (with ), a contradiction with our choice of the sequence .
This leaves us with the second scenario. In this case, by definition of , we can replace by an adequate power with such that the restriction of to some piece of is pseudo-Anosov. However, Daskalopoulos and Wentworth showed that there exists an uniform positive lower bound for
when is pseudo-Anosov on some piece of . Since and is an universal constant, it follows that there exists an uniform positive lower bound for
for all sufficiently large, a contradiction with our choice of the sequences and .
These contradictions show that the sequences and with the properties described above can’t exist.
This completes the proof of the lemma.
3. First derivative of the Weil-Petersson flow
This section concerns the verification of item (VI) of Theorem 1 for the WP flow . More precisely, we will show the following result:
for any and any with
The proof of this result in Burns-Masur-Wilkinson paper is naturally divided into two steps.
In the first step, one shows the following general result providing an estimate for the first derivative of the geodesic flow on arbitrary negatively curved manifold:
Theorem 11 Let be a negatively curved manifold. Consider a geodesic where and suppose that for every the sectional curvatures of any plane containing is greater than for some Lipschitz function .Then,
where is the solution of Riccati equation
with initial data .
We postpone the proof of this theorem to the next post when we will introduce Sasaki metric, Jacobi fields and matrix Riccati equation (among other classical objects) in our way of showing the “abstract” Burns-Masur-Wilkinson ergodicity criterion for geodesic flows.
In the second step, one uses the works of Wolpert to exhibit an adequate bound for the sectional curvatures of the WP metric along WP geodesics . More concretely, one has the following theorem:
- (a) for all ;
- (b) is -controlled in the sense that has a right-derivative satisfying
- (c) ;
- (d) .
Here, denotes the distance between the geodesic segment and .
Proof: Denote by the “WP curvature bound” function provided by Theorem 12 and let be the solution of Riccati’s equation
with initial data .
Since is -controlled (in the sense of item (b) of Theorem 12), it follows that for all : indeed, this is so because , and, if for some , then
Therefore, by applying Theorem 11 in this setting, we deduce that
for and some constant . This completes the proof of Theorem 10.
Closing this post, let us sketch the proof of Theorem 12 (while referring to Subsection 4.4 of Burns-Masur-Wilkinson paper [especially Proposition 4.22 of this article] for more details).
We start by describing how the function is defined. For this sake, we will use Wolpert’s formulas in Theorem 3 above.
More precisely, since the sectional curvatures of the WP metric approach or only near the boundary, we can assume that our geodesic segment in the statement of Theorem 12 is “relatively close” to a boundary stratum , (formally, as Burns-Masur-Wilkinson explain in page 883 of their paper, one must use Proposition 4.7 of their article to produce a nice “thick-thin” decomposition of the Teichmüller space ).
In this setting, for each , we consider the functions and
(where ) along our geodesic segment , . Notice that it is natural to consider these functions in view of the statements in Wolpert’s formulas in Theorem 3.
The WP sectional curvatures of planes containing the tangent vectors to are controlled in terms of and . Indeed, given , we can use a combined length basis to write
Similarly, let us write
By Theorem 3, we obtain the following facts. Firstly, since and are WP-unit vectors, the coefficients are
Secondly, by definition of , we have that
In summary, Wolpert’s formulas (Theorem 3) imply that
(cf. Lemma 4.17 in Burns-Masur-Wilkinson paper).
Now, we want convert the expressions into a positive Lipschitz function satisfying the properties described in items (b), (c), and (d) of Theorem 12, i.e., a -controlled function with appropriately bounded total integral and values at and . We will not give full details on this (and we refer the curious reader to Subsection 4.4 of Burns-Masur-Wilkinson paper), but, as it turns out, the function
where is the (unique) time with for all and is a sufficiently large constant satisfies the conditions in items (a), (b), (c) and (d) of Theorem 12. Here, the basic idea is these properties are consequences of the features of two ODE’s (cf. Lemmas 4.15 and 4.16 in Burns-Masur-Wilkinson paper) for and . For instance, the verification of item (a) (i.e., the fact that controls certain WP sectional curvatures along ) relies on the fact that these two ODE’s permit to prove that
for some sufficiently large constant . In particular, by plugging this into (1), we obtain that
i.e.,, the estimate required by item (a) of Theorem 12.
Concluding this sketch of proof of Theorem 12, let us indicate the two ODE’s on and .
Lemma 13 (Lemma 4.15 of Burns-Masur-Wilkinson paper) .
Proof: By differentiating , we see that
Here, we used the fact that the WP metric is Kähler, so that is parallel (“commutes with ”).
Now, we observe that, by Wolpert’s formulas (cf. Theorem 3), one can write and that
Since (by definition), we conclude from the previous equations that
This proves the lemma.
Remark 15 This ODE is an analogue for the WP metric of Clairaut’s relation for the “model metric” on the surface of revolution of the profil .
Lemma 16 (Lemma 4.15 of Burns-Masur-Wilkinson paper)
Proof: By definition, , so that
Differentiating this equality and using Wolpert’s formulas (Theorem 3), we see that
(Here, we used in the first equality the fact that is a geodesic, i.e., .)
It follows that
This proves the lemma.