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
.
Theorem 3 (Wolpert) Fix
. Then, for any
, and any
and
, the following estimates hold uniformly on
where
is the Kronecker delta.
and, furthermore,
extends continuosly to the boundary stratum
.
- the distance from
to the boundary stratum
is
- for any vector
,
and
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 :
Proposition 4 There exists an universal constant
such that, for any
, one has a holomorphic embedding
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:
Lemma 5 There exists an universal constant
such that, for any
and any cotangent vector
, one has
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:
and
for all
,
and
,
.
Proof: The first inequality
follows from Proposition 4 and Lemma 5. Indeed, by letting , we see from Proposition 4 and Lemma 5 that
Since
we deduce that
i.e., .
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.
The estimates in Proposition 6 (controlling the WP metric in the local coordinates constructed in Proposition 4) permit to deduce the remaining “2/3 of item (IV)” for the WP metric:
Theorem 7 (Burns-Masur-Wilkinson) There are constants
and
such that, for any
, the curvature tensor
of the WP metric satisfies
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
or
in Einstein summation convention, and, in terms of the Christoffel symbols, the coefficients of the curvature tensor 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
and, consequently,
This completes the proof.
At this point, we have that Theorems 3 and 7 imply the validity of item (IV) of Burns-Masur-Wilkinson ergodicity criterion (Theorem 1) for the WP metric.
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.,
Theorem 8 There exists a constant
such that for all
,
, one has the following polynomial lower bound on the injectivity radius of the WP metric at
:
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:
Theorem 10 There are constants
,
,
and
such that
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:
Theorem 12 There are constants
and
such that for any
and any geodesic segment
there exists a positive Lipschitz function
with
- (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
.
Using Theorems 11 and 12, we can easily complete the proof of Theorem 10 (i.e., the verification of item (VI) of Burns-Masur-Wilkinson ergodicity criterion for the WP metric):
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
Finally,
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
and
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.
Reblogged this on Weblog.
By: Pengfei on July 2, 2015
at 5:12 am