Posted by: matheuscmss | December 31, 2013

Dynamics of the Weil-Petersson flow: basic geometry of the Weil-Petersson metric II

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 {N} be the quotient {N=M/\Gamma} of a contractible, negatively curved, possibly incomplete, Riemannian manifold {M} by a subgroup {\Gamma} of isometries of {M} acting freely and properly discontinuously. Denote by {\overline{N}} the metric completion of {N} and {\partial N:=\overline{N}-N} the boundary of {N}.Suppose that:

  • (I) the universal cover {M} of {N} is geodesically convex, i.e., for every {p,q\in M}, there exists an unique geodesic segment in {M} connecting {p} and {q}.
  • (II) the metric completion {\overline{N}} of {(N,d)} is compact.
  • (III) the boundary {\partial N} is volumetrically cusplike, i.e., for some constants {C>1} and {\nu>0}, the volume of a {\rho}-neighborhood of the boundary satisfies

    \displaystyle \textrm{Vol}(\{x\in N: d(x,\partial N)<\rho\})\leq C \rho^{2+\nu}

    for every {\rho>0}.

  • (IV) {N} has polynomially controlled curvature, i.e., there are constants {C>1} and {\beta>0} such that the curvature tensor {R} of {N} and its first two derivatives satisfy the following polynomial bound

    \displaystyle \max\{\|R(x)\|,\|\nabla R(x)\|,\|\nabla^2 R(x)\|\}\leq C d(x,\partial N)^{-\beta}

    for every {x\in N}.

  • (V) {N} has polynomially controlled injectivity radius, i.e., there are constants {C>1} and {\beta>0} such that

    \displaystyle \textrm{inj}(x)\geq (1/C) d(x,\partial N)^{\beta}

    for every {x\in N} (where {inj(x)} denotes the injectivity radius at {x}).

  • (VI) The first derivative of the geodesic flow {\varphi_t} is polynomially controlled, i.e., there are constants {C>1} and {\beta>0} such that, for every infinite geodesic {\gamma} on {N} and every {t\in [0,1]}:

    \displaystyle \|D_{\stackrel{.}{\gamma}(0)}\varphi_t\|\leq C d(\gamma([-t,t]),\partial N)^{\beta}

Then, the Liouville (volume) measure {m} of {N} is finite, the geodesic flow {\varphi_t} on the unit cotangent bundle {T^1N} of {N} is defined at {m}-almost every point for all time {t}, and the geodesic flow {\varphi_t} is non-uniformly hyperbolic (in the sense of Pesin’s theory) and ergodic.

Actually, the geodesic flow {\varphi_t} is Bernoulli and, furthermore, its metric entropy {h(\varphi_t)} is positive, finite and {h(\varphi_t)} is given by Pesin’s entropy formula (i.e., it is equal to the sum of positive Lyapunov exponents of {\varphi_t} 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 {\partial \mathcal{T}} of the Teichmüller space {\mathcal{T}=Teich(S)}.

Before stating Wolpert’s formulas, we need an adapted system of coordinates (called combined length basis in the literature) near the strata {\mathcal{T}_{\sigma}}, {\sigma\in\mathcal{C}(S)}, of {\partial\mathcal{T}}, where {\mathcal{C}(S)} is the curve complex of {S} (introduced in the previous post).

Denote by {\mathcal{B}} the set of pairs (“basis”) {(\sigma,\chi)} where {\sigma\in\mathcal{C}(S)} is a simplex of the curve complex and {\chi} is a collection of simple closed curves such that each {\beta\in\chi} is disjoint from all {\alpha\in\sigma}. Here, we allow that two curves {\beta, \beta'\in\chi} intersect (i.e., one might have {\beta\cap\beta'\neq\emptyset}) and also the case {\chi=\emptyset} is not excluded.

Following the nomenclature introduced by Wolpert, we say that {(\sigma,\chi)\in\mathcal{B}} is a combined length basis at a point {X\in\mathcal{T}} whenever the set of tangent vectors

\displaystyle \{\lambda_{\alpha}(X), J\lambda_{\alpha}(X), \textrm{grad}\ell_{\beta}(X)\}_{\alpha\in\sigma, \beta\in\chi}

is a basis of {T_X\mathcal{T}}, where {\ell_{\gamma}} is the length parameter in the Fenchel-Nielsen coordinates and {\lambda_{\alpha}:=\textrm{grad} \ell_{\alpha}^{1/2}}.

Remark 1 The length parameters {\ell_{\gamma}} and their square-roots {\ell_{\gamma}^{1/2}} 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 {(\sigma,\chi)} as a combination of a collection {\sigma\in\mathcal{C}(S)} of short curves (indicating the boundary stratum that one is close to), and a collection {\chi} of relative curves to {\sigma} allowing to complete the set {\{\lambda_{\alpha}\}_{\alpha\in\sigma}} into a basis of the tangent space to {\mathcal{T}} in which one can write nice formulas for the WP metric.

This notion can be “extended” to a stratum {\mathcal{T}_{\sigma}} of {\mathcal{T}} as follows. We say {\chi} is a relative basis at a point {X_{\sigma}\in\mathcal{T}_{\sigma}} whenever {(\sigma,\chi)\in\mathcal{B}} and the length parameters {\{\ell_{\beta}\}_{\beta\in\chi}} is a local system of coordinates for {\mathcal{T}_{\sigma}} near {X_{\sigma}}.

Remark 2 The stratum {\mathcal{T}_{\sigma}} is (isomorphic to) a product of the Teichmüller spaces of the pieces of {X_{\sigma}\in\mathcal{T}_{\sigma}}. In particular, {\mathcal{T}_{\sigma}} carries a “WP metric”, namely, the product of the WP metrics on the Teichmüller spaces of the pieces of {X_{\sigma}}. In this setting, {\chi} is a relative basis at {X_{\sigma}\in\mathcal{T}_{\sigma}} if and only if {\{\textrm{grad} \ell_{\beta}\}_{\beta\in\chi}} is a basis of {T_{X_{\sigma}}\mathcal{T}_{\sigma}}.

Remark 3 Contrary to the Fenchel-Nielsen coordinates, the length parameters {\{\ell_{\beta}\}_{\beta\in\chi}} associated to a relative basis {\chi} might not be a global system of coordinates for {\mathcal{T}_{\sigma}}. Indeed, this is so because we allow the curves in {\chi} to intersect non-trivially: geometrically, this means that there are points {X_0} in {\mathcal{T}_{\sigma}} where the geodesic representatives of such curves meet orthogonally, and, at such points {X_0}, the system of coordinates induced by {\{\ell_{\beta}\}_{\beta\in\chi}} 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 {X_{\sigma}\in\mathcal{T}_{\sigma}}, {\sigma\in\mathcal{C}(S)} , there exists a relative length basis {\chi}. Furthermore, the WP metric {\langle.,.\rangle_{WP}} can be written as

\displaystyle \langle.,.\rangle_{WP}\sim \sum\limits_{\alpha\in\sigma}\left((d\ell_{\alpha}^{1/2})^2+(d\ell_{\alpha}^{1/2}\circ J)^2\right) + \sum\limits_{\beta\in\chi}(d\ell_{\beta})^2

where the implied comparison constant is uniform in a neighborhood {U\subset\overline{\mathcal{T}}} of {X_{\sigma}}. In particular, there exists a neighborhood {V\subset\overline{\mathcal{T}}} of {X_{\sigma}} such that {(\sigma,\chi)} is a combined length basis at any {X\in V\cap\mathcal{T}}.

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 {(\sigma,\chi)}.

In order to write down the next list of formulas of Wolpert, we need the following notations. Given {\mu} an arbitrary collection of simple closed curves on {S}, we define

\displaystyle \underline{\ell}_{\mu}(X):=\min\limits_{\alpha\in\mu}\ell_{\alpha}(X)\quad \textrm{and}\quad \overline{\ell}_{\mu}(X):=\max\limits_{\alpha\in\mu}\ell_{\alpha}(X)

where {X\in\mathcal{T}=Teich(S)}. Also, given a constant {c>1} and a basis {(\sigma,\chi)\in\mathcal{B}}, we will consider the following (Bers) region of Teichmüller space:

\displaystyle \Omega(\sigma,\chi,c):=\{X\in\mathcal{T}: 1/c<\underline{\ell}_{\chi}(X) \textrm{ and } \overline{\ell}_{\sigma\cup\chi}(X)<c\}

Wolpert provides several estimates for the WP metric {\langle.,.\rangle_{WP}=\langle.,.\rangle} and its sectional curvatures in terms of the basis {\lambda_{\alpha}=\textrm{grad}\ell_{\alpha}^{1/2}}, {\alpha\in\sigma} and {\textrm{grad}\ell_{\beta}}, {\beta\in\chi}, which are uniform on the regions {\Omega(\sigma,\chi,c)}.

Theorem 3 (Wolpert) Fix {c>1}. Then, for any {(\sigma,\chi)\in\mathcal{B}}, and any {\alpha,\alpha'\in\sigma} and {\beta,\beta'\in\chi}, the following estimates hold uniformly on {\Omega(\sigma,\chi,c)}

  • {\langle\lambda_{\alpha},\lambda_{\alpha'}\rangle = \frac{1}{2\pi}\delta_{\alpha,\alpha'}+O((\ell_{\alpha}\ell_{\alpha'})^{3/2}) = \langle J\lambda_{\alpha}, J\lambda_{\alpha'}\rangle} where {\delta_{\ast,\ast\ast}} is the Kronecker delta.
  • {\langle\lambda_{\alpha}, J\lambda_{\alpha'}\rangle=\langle J\lambda_{\alpha}, \textrm{grad} \ell_{\beta}\rangle=0}
  • {\langle\textrm{grad} \ell_{\beta}, \textrm{grad} \ell_{\beta'}\rangle\sim 1} and, furthermore, {\langle\textrm{grad} \ell_{\beta}, \textrm{grad} \ell_{\beta'}\rangle} extends continuosly to the boundary stratum {\mathcal{T}_{\sigma}}.
  • {\langle\lambda_{\alpha},\textrm{grad}\ell_{\beta}\rangle = O(\ell_{\alpha}^{3/2})}
  • the distance from {X\in\Omega(\sigma,\chi,c)} to the boundary stratum {\mathcal{T}_{\sigma}} is

    \displaystyle d(X,\mathcal{T}_{\sigma}) = \sqrt{2\pi\sum\limits_{\alpha\in\sigma}\ell_{\alpha}(X)} + O\left(\sum\limits_{\alpha\in\sigma}\ell_{\alpha}^{5/2}(X)\right)

  • for any vector {v\in T\Omega(\sigma,\chi,c)},

    \displaystyle \left\|\nabla_v \lambda_{\alpha} - \frac{3}{2\pi\ell_{\alpha}^{1/2}}\langle v, J\lambda_{\alpha}\rangle J\lambda_{\alpha}\right\|_{WP} = O(\ell_{\alpha}^{3/2}\|v\|_{WP})

  • {\|\nabla_{\lambda_{\alpha}}\textrm{grad} \ell_{\beta}\|_{WP} = O(\ell_{\alpha}^{1/2})} and {\|\nabla_{\lambda_{\alpha}}\textrm{grad} \ell_{\beta}\|_{WP}=O(\ell_{\alpha}^{1/2})}
  • {\nabla_{\textrm{grad}\ell_{\beta}}\textrm{grad}\ell_{\beta'}} extends continuously to the boundary stratum {\mathcal{T}_{\sigma}}
  • the sectional curvature of the complex line (real two-plane) {\{\lambda_{\alpha}, J\lambda_{\alpha}\}} is

    \displaystyle \langle R(\lambda_{\alpha}, J\lambda_{\alpha})J\lambda_{\alpha}, \lambda_{\alpha}\rangle = \frac{3}{16\pi^2\ell_{\alpha}} + O(\ell_{\alpha})

  • for any quadruple {(v_1, v_2, v_3, v_4)}, {v_i\in\{\lambda_{\alpha}, J\lambda_{\alpha}, \textrm{grad}\ell_{\beta}\}_{\alpha\in\sigma, \beta\in\chi}} distinct from a curvature-preserving permutation of {(\lambda_{\alpha}, J\lambda_{\alpha}, J\lambda_{\alpha}, \lambda_{\alpha})}, one has

    \displaystyle \langle R(v_1,v_2)v_3, v_4\rangle=O(1),

    and, moreover, each {v_i} of the form {\lambda_{\alpha}} or {J\lambda_{\alpha}} introduces a multiplicative factor {O(\ell_{\alpha})} 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 {\{\lambda_{\alpha}, J\lambda_{\alpha}\}} converge to {-\infty} with speed {\sim -1/\ell_{\alpha}\sim -1/d(X,\mathcal{T}_{\sigma})^2} as one approaches the boundary stratum {\mathcal{T}_{\sigma}}, while the sectional curvatures of the WP metric associated to quadruples of the form {(\lambda_{\alpha}, J\lambda_{\alpha}, J\lambda_{\alpha'}, \lambda_{\alpha'})} with {\alpha, \alpha'\in\sigma} and {\alpha\neq\alpha'} converge to {0} with speed {\sim O(\ell_{\alpha}^2 \ell_{\alpha'}^2) \sim O(d(X,\mathcal{T}_{\sigma})^8)} at least.

In particular, these formulas of Wolpert allow to show “1/3 of item (IV)” for the WP metric, that is,

\displaystyle \|R_{WP}(x)\|_{WP}\leq C d(x,\partial\mathcal{T})^{-2}

for all {x\in\mathcal{T}}.

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 {-\infty}, 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 {Teich(S)} approaching zero whenever {\sigma\in\mathcal{C}(S)} contains two distinct curves. In other words, the WP metric has sectional curvatures approaching zero whenever the genus {g} and the number of punctures {n} of {S=S_{g,n}} satisfy {3g-3+n>1}, i.e., except in the cases of once-punctured torii {S_{1,1}} and four-times puncture spheres {S_{0,4}}. This qualitative difference on the geometry of the WP metric on {Teich_{g,n}} in the cases {3g-3+n>1} and {3g-3+n=1} (i.e., {(g,n)=(0,4)} or {(1,1)}) 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 {\mathcal{M}_{1,1}\simeq\mathbb{H}^2/PSL(2,\mathbb{Z})} in a {\varepsilon}-neighborhood of the cusp at infinity as a {C^2}-pertubation of the metric {\pi^3(4dr^2+r^6d\theta)} of the surface of revolution of the profile {\{y=x^3\}} modulo multiplicative factors of the form {1+O(r^4)}.

Now, we will investigate the remaining “2/3 of item (IV)” for the WP metric, i.e., polynomial bounds for the first two derivatives {\nabla R} and {\nabla^2 R} of the curvature operator {R} 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 {C^0}-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 {Teich(S)} 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 {Teich(S)}.

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 {Teich(S)} as a totally real submanifold of the so-called quasi-Fuchsian locus {QF(S)}, and, in this setting, the Weil-Petersson symplectic {2}-form {\omega_{WP}} is the restriction to {Teich(S)} of the differential of a holomorphic {1}-form {\theta_{WP}} globally defined on the quasi-Fuchsian locus {QF(S)}. In particular, it is possible to use Cauchy’s inequalities to the holomorphic object {\theta_{WP}} 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 {f:\mathbb{R}\rightarrow\mathbb{C}} (“WP metric”) knowing some bounds for the values of {f}. In principle, we can not do this by simply applying Cauchy’s estimates to {f}, but in our context we know (“by the results of McMullen”) that the natural embedding {\mathbb{R}\subset \mathbb{C}=\mathbb{R}\oplus i\mathbb{R}} of {\mathbb{R}} as a totally real submanifold of {\mathbb{C}} allows to think of {f:\mathbb{R}\rightarrow\mathbb{C}} as the restriction of a holomorphic function {g:\mathbb{C}\rightarrow\mathbb{C}} and, thus, we can apply Cauchy inequalities to {g} to get some estimates for {f}.

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 {Teich(S)} into the quasi-Fuchsian locus {QF(S)} and the holomorphic {1}-form {\theta_{WP}} of McMullen whose differential restricts to the WP symplectic {2}-form on {Teich(S)}. 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 {QF(S)} and McMullen’s {1}-forms {\theta_{WP}}

Given a hyperbolic Riemann surface {S=\mathbb{H}/\Gamma}, {\Gamma<PSL(2,\mathbb{R})}, the quasi-Fuchsian locus {QF(S)} is defined as

\displaystyle QF(S) = Teich(S)\times Teich(\overline{S})

where {\overline{S}} is the conjugate Riemann surface of {S}, i.e., {\overline{S}} is the quotient {\overline{S}=\mathbb{L}/\Gamma} of the lower-half plane {\mathbb{L}=\{z\in\mathbb{C}:\textrm{Im}(z)<0\}} by {\Gamma}. The Fuchsian locus {F(S)} is the image of {Teich(S)} under the anti-diagonal embedding

\displaystyle \widehat{\alpha}:Teich(S)\rightarrow QF(S), \quad \widehat{\alpha}(X)=(X,\overline{X})

Geometrically, we can think of elements {(X,Y)\in QF(S)} as follows. Recall that {X} and {Y} are related to {S} and {\overline{S}} via (extremal) quasiconformal mappings determined by the solutions of Beltrami equations associated to {\Gamma}-invariant Beltrami differentials (coefficients) {\mu_X} and {\mu_{Y}} on {\mathbb{H}} and {\mathbb{L}}. Now, we observe that {\mathbb{H}} and {\mathbb{L}} live naturally on the Riemann sphere {\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}}. Since the real axis/circle at infinity/equator {\mathbb{R}_{\infty}=\overline{\mathbb{C}}-(\mathbb{H}\cup\mathbb{L})} has zero Lebesgue measure, we see that {\mu_X} and {\mu_Y} induce a Beltrami differential {\mu_{(X,Y)}} on {\overline{\mathbb{C}}}. By solving the corresponding Beltrami equation, we obtain a quasiconformal map {f_{X,Y}} on {\overline{\mathbb{C}}} and, by conjugating, we obtain a quasi-Fuchsian subgroup

\displaystyle \Gamma(X,Y)=\{f_{(X,Y)}\circ\gamma\circ f_{(X,Y)}^{-1}: \gamma\in\Gamma\}<PSL(2,\mathbb{C}),

i.e., a Kleinian subgroup whose domain of discontinuity {\Omega(X,Y)\subset\overline{\mathbb{C}}} consists of two connected components {A} and {B} such that {X\simeq A/\Gamma(X,Y)} and {Y\simeq B/\Gamma(X,Y)}.

The following picture summarizes the discussion of the previous paragraph:

WP-2013-3-1

Remark 8 The Jordan curve given by the image {f_{(X,Y)}(\mathbb{R}_{\infty})} of the equator {\mathbb{R}_{\infty}} under the quasiconformal map {f_{(X,Y)}} is “wild” in general, e.g., it has Hausdorff dimension {>1} (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 {\Gamma(X,Y)} attached to {(X,Y)\in QF(S)=Teich(S)\times Teich(\overline{S})} permits to assign (marked) projective structures to {X} and {Y}. More precisely, by writing {X\simeq A/\Gamma(X,Y)} and {Y\simeq B/\Gamma(X,Y)} with {A, B\subset\overline{\mathbb{C}}} and {\Gamma(X,Y)<PSL(2,\mathbb{C})}, we are equipping {X} and {Y} with projective structures, that is, atlases of charts to {\mathbb{C}} whose changes of coordinates are Möebius transformations (i.e., elements of {PSL(2,\mathbb{C})}). Furthermore, by recalling that {X} and {Y} come with markings {f:S\rightarrow X} and {g:\overline{S}\rightarrow Y} (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

\displaystyle \sigma:QF(S)\rightarrow Proj(S)\times Proj(\overline{S})

assigning to {(X,Y)} the marked projective structures

\displaystyle \sigma(X,Y):=(\sigma_{QF}(X,Y),\overline{\sigma}_{QF}(X,Y))

Here, {Proj(S)} is the “Teichmüller space of projective structures” on {S}, i.e., the space of “Teichmüller” equivalence classes of marked projective structures {f:S\rightarrow X} where two marked projective structures {f_1:S\rightarrow X_1} and {f_2:S\rightarrow X_2} are “Teichmüller” equivalent whenever there is a projective isomorphism {h:X_1\rightarrow X_2} homotopic to {f_2\circ f_1^{-1}}.

Remark 9 The procedure (due to Bers) of attaching a quasi-Fuchsian subgroup {\Gamma(X,Y)} to a pair of hyperbolic surfaces {X} and {Y} is called Bers simultaneous uniformization because the knowledge of {\Gamma(X,Y)} allows to equip at the same time {X} and {Y} with natural projective structures.

Note that {\sigma} is a section of the natural projection

\displaystyle Proj(S)\times Proj(\overline{S})\rightarrow QF(S)=Teich(S)\times Teich(\overline{S})

obtained by sending each pair of (marked) projective structures {(X,Y)}, {X\in Proj(S)}, {Y\in Proj(\overline{S})}, to the unique pair of (marked) compatible conformal structures {(\pi(X), \overline{\pi}(Y))}, {\pi(X)\in Teich(S)}, {\overline{\pi}(Y)\in Teich(\overline{S})}.

We will now describe how the (affine) structure of the fibers {Proj_X(S)=\pi^{-1}(X)} of the projection {\pi: Proj(S)\rightarrow Teich(S)} and the section {\sigma} can be used to construct McMullen’s primitives/potentials of the Weil-Petersson symplectic form {\omega_{WP}}.

Given two projective structures {p_1, p_2\in Proj_X(S)} in the same of the projection {\pi: Proj(S)\rightarrow Teich(S)}, one can measure how far apart from each other are {p_1} and {p_2} using the so-called Schwarzian derivative.

More precisely, the fact that {p_1} and {p_2} induce the same conformal structure means that the charts of atlases associated to them can be thought as some families of maps {f_1:U\rightarrow\overline{\mathbb{C}}} and {f_2:U\rightarrow\overline{\mathbb{C}}} from (small) open subsets {U\subset X} to the Riemann sphere {\overline{\mathbb{C}}}, and we can measure the “difference” {p_2-p_1} by computing how “far” from a Möebius transformation (element of {PSL(2,\mathbb{C})}) is {f_2\circ f_1^{-1}}.

Here, given a point {z\in U}, one observes that there exists an unique Möebius transformation {A\in PSL(2,\mathbb{C})} such that {f_2} and {A\circ f_1} coincide at {z} up to second order (i.e., {f_2} and {A\circ f_1} have the same value and the same first and second derivatives at {z}). Hence, it is natural to measure how far from a Möebius transformation is {f_2\circ f_1^{-1}} by understanding the difference between the third derivatives of {f_2} and {A\circ f_1} at {z\in U}, i.e., {D^3(f_2-A\circ f_1)(z)}.

Actually, this is almost the definition of the Schwarzian derivative: since the derivatives of {f_2} and {A\circ f_1} map {T_z U} to {T_{f_2(z)}\overline{\mathbb{C}}}, in order to recover an object from {T_zU} to itself, it is a better idea to “correct” {D^3(f_2-A\circ f_1)(z)} with {Df_2^{-1}(z)}, i.e., we define the Schwarzian derivative {S\{f_2,f_1\}(z)} of {f_2} and {f_1} at {z} as

\displaystyle S\{f_2,f_1\}(z):=6\left(Df(z)^{-1}\circ D^3(f_2-A\circ f_1)(z)\right)

Here, the factor {6} shows up for historical reasons (that is, this factor makes {S\{f_2,f_1\}(z)} coincide with the classical definition of Schwarzian derivative in the literature).

By definition, the Schwarzian derivative {S\{f_2, f_1\}} is a field of quadratic forms on {U} (since its definition involves taking third order derivatives). In other terms, {S\{f_2, f_1\}} is a quadratic differential on {U}, that is, the “difference” {p_2-p_1} between two projective structures {p_1, p_2\in Proj_X(S)} in the same fiber of the projection {\pi:Proj(S)\rightarrow Teich(S)} is given by a quadratic differential {p_2-p_1=S\{p_2, p_1\}\in Q(X)}. In particular, the fibers {Proj_X(S)} are affine spaces modeled by the space {Q(X)} of quadratic differentials on {X}.

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 {Teich(S)} by computing the eccentricities of quasiconformal maps between these points.

Using this affine structure on {Proj_X(S)} and the fact that {Q(X)\simeq T_X^*Teich(S)} is the cotangent space of {Teich(S)} at {X}, we see that, for each {Y, Z\in Teich(\overline{S})}, the map

\displaystyle X\in Teich(S)\mapsto \sigma_{QF}(X,Y)-\sigma_{QF}(X,Z)\in Q(X)

defines a (holomorphic) {1}-form on {Teich(S)}. Note that, by letting {Y\in Teich(\overline{S})} vary and by fixing {Z\in Teich(\overline{S})}, we have a map {\tau_Z=\tau} given by

\displaystyle (X,Y)\in Teich(S)\times Teich(\overline{S})\mapsto \tau(X,Y):=\sigma_{QF}(X,Y)-\sigma_{QF}(X,Z)\in Q(X)

Since {QF(S)=Teich(S)\times Teich(\overline{S})} (so that {T^*QF(S)=T^*Teich(S)\oplus T^*Teich(\overline{S})}) and {Q(X)\simeq T^*_X Teich(S)}, we can think of {\tau} as a (holomorphic) {1}-form on {QF(S)}.

For later use, let us notice that the {1}-form {\tau:QF(S)\rightarrow T^*Teich(S)} is bounded with respect to the Teichmüller metric on {Teich(S)}. Indeed, this is a consequence of Nehari’s bound stating that if {U\subset\overline{\mathbb{C}}} is a round disc (i.e., the image of the unit disc {\mathbb{D}\subset \mathbb{C}\subset\overline{\mathbb{C}}} under a Möebius transformation) equipped with its hyperbolic metric {\rho} and {f:U\rightarrow\mathbb{C}} is an injective complex-analytic map, then

\displaystyle \|S\{f,z\}\|_{L^{\infty}}\leq 3/2.

In this setting, McMullen constructed primitives/potentials for the WP symplectic form {\omega_{WP}} as follows. The Teichmüller space {Teich(S)} sits in the quasi-Fuchsian locus {QF(S)} as the Fuchsian locus {F(S)=\widehat{\alpha}(Teich(S))} where {\widehat{\alpha}} is the anti-diagonal embedding

\displaystyle \widehat{\alpha}:Teich(S)\rightarrow QF(S), \quad \widehat{\alpha}(X)=(X,\overline{X})

By pulling back the {1}-form {\tau} under {\widehat{\alpha}}, we obtain a bounded {1}-form

\displaystyle \theta_{WP}(X):=\widehat{\alpha}^*(\tau)(X)=\sigma_{QF}(X,\overline{X})-\sigma_{QF}(X,Z)

Remark 12 This form {\theta_{WP}=\widehat{\alpha}^*(\tau)} is closely related to a classical object in Teichmüller theory called Bers embedding: in our notation, the Bers embedding is

\displaystyle \beta_X(\overline{Z})=\sigma_{QF}(X,Z)-\sigma_{QF}(X,\overline{X})=-\widehat{\alpha}^*(\tau)(X)=-\theta_{WP}(X)

McMullen showed that the bounded {1}-forms {i\theta_{WP}} are primitives/potentials of the WP symplectic {2}-form {\omega_{WP}}, i.e.,

\displaystyle d(i\theta_{WP})=\omega_{WP}

See also Section 7.7 of Hubbard’s book for a nice exposition of this theorem of McMullen. Equivalently, the restriction of the holomorphic {1}-form {\tau} to the Fuchsian locus {F(S)} (a totally real sublocus of {QF(S)}) permits to construct (Teichmüller bounded) primitives for the WP symplectic form on {F(S)}.

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 {\omega_{WP}} after Burns-Masur-Wilkinson

Following Burns-Masur-Wilkinson, we will need the following local coordinates in {Teich(S)}:

Proposition 4 There exists an universal constant {C_0=C_0(g,n)\geq 1} such that, for any {X_0\in Teich(S)=Teich_{g,n}}, one has a holomorphic embedding

\displaystyle \psi=\psi_{X_0}:\Delta^N\rightarrow Teich(S)

of the Euclidean unit polydisc {\Delta^N:=\{(z_1,\dots,z_N)\in \mathbb{C}^N: |z_j|<1\,\,\,\,\forall\,j=1,\dots,N\}} (where {N=3g-3+n=\textrm{dim}(Teich(S))}) sending {0\in\Delta^N} to {X_0=\psi(0)} and satisfying

\displaystyle \frac{1}{C_0}\|v\|\leq \|D\psi(v)\|_T\leq C_0\|v\|, \quad\forall v\in T\Delta^N,

where {\|.\|_T} is the Teichmüller norm and {\|.\|} is the Euclidean norm on {\Delta^N}.

This result is proven in this paper of McMullen here.

Also, since the statement of Proposition 4 involves the Teichmüller norm {\|.\|_T} and we are interested in the Weil-Petersson norm {\|.\|_{WP}}, the following comparison (from Lemma 5.4 of Burns-Masur-Wilkinson paper) between {\|.\|_T} and {\|.\|_{WP}} will be helpful:

Lemma 5 There exists an universal constant {C=C(g,n)\geq 1} such that, for any {X\in Teich(S)} and any cotangent vector {\phi\in Q(X)\simeq T_X^*Teich(S)}, one has

\displaystyle \|\phi\|_{WP}\leq C\frac{1}{\underline{\ell}(X)}\|\phi\|_T

where {\underline{\ell}(X)} is the systole of {X}(i.e., the length of the shortest closed simple hyperbolic geodesics on {X}). In particular, for any {X\in Teich(S)} and any tangent vector {\mu\in T_X Teich(S)}, one has

\displaystyle \|\mu\|_{T}\leq C\frac{1}{\underline{\ell}(X)}\|\mu\|_{WP}

Proof: Given {X\in Teich(S)}, let us write {X\simeq \mathbb{H}^2/\Gamma} with {\Gamma<PSL(2,\mathbb{R})} is “normalized” to contain the element {T(z)=\lambda z} where {\lambda=\log\underline{\ell}(X)}.

Fix {D\subset\mathbb{H}} a Dirichlet fundamental domain of the action of {\Gamma} centered at the point {i\in\mathbb{H}}.

By the collaring theorem stating that a closed simple hyperbolic geodesic {\gamma} of length {\ell} has a collar {A(\gamma,\eta(\ell))} [tubular neighborhood] of radius {\eta(\ell):=(1/2)\log((\cosh(\ell/2)+1)/(\cosh(\ell/2)-1))} isometrically embedded in {X} and two of these collars {A(\gamma_1,\eta(\ell_1))} and {A(\gamma_2,\eta(\ell_2))} are disjoint whenever {\gamma_1} and {\gamma_2} are disjoint (see, e.g., Theorem 3.8.3 in Hubbard’s book), we have that the union of {1/\underline{\ell}(X)} isometric copies of {D} contains a ball {B} of fixed (universal) radius {c=c(g,n)>0} around any point {z\in D}.

By combining the Cauchy integral formula with the fact stated in the previous paragraph, we see that

\displaystyle |\phi(z)|\leq \frac{1}{2\pi c}\int_B |\phi|\leq \frac{1}{2\pi c\underline{\ell}(X)} \int_D |\phi|=\frac{1}{2\pi c\underline{\ell}(X)}\|\phi\|_T

Since the hyperbolic metric {\rho} is bounded away from {0} on {D}, we can use the {L^{\infty}}-norm estimate on {\phi} above to deduce that

\displaystyle \|\phi\|_{WP}^2:=\int_D\frac{|\phi|^2}{\rho^2} \leq \frac{C}{\underline{\ell}(X)^2}\|\phi\|_T^2

for some constant {C=C(g,n)>0}. This completes the proof of the lemma. \Box

Remark 13 The factor {1/\underline{\ell}(X)} in the previous lemma can be replaced by {1/\sqrt{\underline{\ell}(X)}} 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 “{g_{ij}}‘s” of the WP metric. More concretely, denoting by {\psi=\psi_{X_0}} “centered at some {X_0\in Teich(S)}” in Proposition 4, let {z_k=x_k+iy_k}, {k=1,\dots, N} and consider the vector fields

\displaystyle e_{\ell}:=\left\{\begin{array}{cl} \partial/\partial x_{\ell}, & \textrm{if } \ell=1,\dots, N \\ \partial/\partial y_{\ell-N}, & \textrm{if } \ell=N+1,\dots, 2N\end{array}\right.

on {\Delta^N}. In setting, we denote by {G_{ij}(z)=\psi^*g_{WP}(z)(e_i,e_j)} the “{g_{ij}}‘s” of the WP metric {g_{WP}} in the local coordinate {\psi} and by {G^{-1}(z)=(G^{ij}(z))_{1\leq i,j\leq 2N}} the inverse of the matrix {(G_{ij}(z))_{1\leq i,j\leq 2N}}.

Proposition 6 There exists an universal constant {C=C(g,n)\geq 1} such that, for any {X_0\in Teich(S)}, the pullback {G=\psi^*g_{WP}} of the WP metric {g_{WP}} local coordinate {\psi=\psi_{X_0}:\Delta^N\rightarrow Teich(S)} “centered at {X_0}” in Proposition 4 verifies the following estimates:

\displaystyle \|G^{-1}(z)\|\leq C/\underline{\ell}(X_0)^2 \quad \forall z\in\Delta^N, \, \|z\|<1/2,

and

\displaystyle \max\limits_{(\xi_1,\dots,\xi_k)\in\{x_1,\dots, x_N, y_1,\dots, y_N\}^k}\frac{1}{k!}\left|\frac{\partial^k G_{ij}}{\partial\xi_1\dots\partial\xi_k}(z)\right|\leq C

for all {1\leq i,j\leq 2N}, {k\geq 0} and {z\in\Delta^N}, {\|z\|<1/2}.

Proof: The first inequality

\displaystyle \|G^{-1}(z)\|\leq C/\underline{\ell}(X_0)^2

follows from Proposition 4 and Lemma 5. Indeed, by letting {v=\sum\limits_{i=1}^{2N} v_i e_i}, we see from Proposition 4 and Lemma 5 that

\displaystyle \|v\|^2\leq C_0^2\|D\psi(v)\|_T^2\leq \frac{C}{\underline{\ell}(X_0)^2}\|D\psi(v)\|_{WP}^2

Since

\displaystyle \begin{array}{rcl} \|D\psi(v)\|_{WP}^2&=&\langle D\psi(v), D\psi(v)\rangle_{WP}=\sum v_i v_j\langle D\psi(e_i), D\psi(e_j)\rangle_{WP} \\ &=& \sum v_i v_j G_{ij}=\langle v, Gv\rangle \\ &\leq& \|v\|\cdot\|Gv\|, \end{array}

we deduce that

\displaystyle \|v\|^2\leq \frac{C}{\underline{\ell}(X_0)^2}\|v\|\cdot\|Gv\|,

i.e., {\|G^{-1}\|\leq C/\underline{\ell}(X_0)^2}.

For the proof of second inequality (estimates of the {k}-derivatives of {G_{ij}}‘s), we begin by “rephrasing” the construction of McMullen’s {\theta_{WP}}-form in terms of the local coordinate {\psi=\psi_{X_0}} introduced in Proposition 4.

The composition {\widehat{\alpha}\circ\psi} of the local coordinate {\psi:\Delta^N\rightarrow Teich(S)} with the anti-diagonal embedding {\widehat{\alpha}:Teich(S)\rightarrow QF(S)} of the Teichmüller space in the quasi-Fuchsian locus can be rewritten as

\displaystyle \widehat{\alpha}\circ\psi = \Psi\circ \alpha

where {\alpha:\Delta^N\rightarrow\Delta^N\times\Delta^N} is the anti-diagonal embedding

\displaystyle \alpha(z)=(z,\overline{z})

and the local coordinate {\Psi:\Delta^N\times\Delta^N\rightarrow QF(S)} given by

\displaystyle \Psi(z,w)=(\psi(z),\overline{\psi(\overline{w})}).

In this setting, the pullback by {\Psi} of the holomorphic {1}-form {\tau(X,Y)=\sigma_{QF}(X,Y)-\sigma_{QF}(X,Z)} gives a holomorphic {1}-form {\kappa=\Psi^*\tau} on {\Delta^N\times\Delta^N}. Moreover, since the Euclidean metric on {\Delta^N\times\Delta^N} is comparable to the pullback by {\Psi} of the Teichmüller metric (cf. Proposition 4), {\tau} is bounded in Teichmüller metric and {d(i\theta_{WP})=\omega_{WP}} where {\theta_{WP}=\widehat{\alpha}^*\tau}, we see that

\displaystyle \alpha^*\Omega=\psi^*\omega_{WP}

where {\Omega:=d(i\kappa)} and {\kappa:=\Psi^*\tau} is a holomorphic bounded (in the Euclidean norm) {1}-form on {\Delta^N\times\Delta^N}.

Let us write {\kappa=\sum\limits_{j=1}^N a_j dz_j} in complex coordinates {(z_1,\dots,z_N,w_1,\dots,w_N)\in\Delta^N\times\Delta^N}, where {a_j:\Delta^N\times\Delta^N\rightarrow\mathbb{C}} are bounded holomorphic functions. Hence,

\displaystyle \Omega=d(i\kappa)=i\left(\sum\limits_{j,k=1}\frac{\partial a_j}{\partial z_k} dz_k\wedge dz_j + \sum\limits_{j,k=1}\frac{\partial a_j}{\partial w_k} dw_k\wedge dz_j\right)

and, a fortiori,

\displaystyle \psi^*\omega_{WP}=\alpha^*\Omega=i\left(\sum\limits_{j,k=1}\frac{\partial a_j}{\partial z_k} dz_k\wedge dz_j + \sum\limits_{j,k=1}\frac{\partial a_j}{\partial \overline{z}_k} d\overline{z}_k\wedge dz_j\right)

Since {\psi^*\omega_{WP}} is the Kähler form of the metric {G=\psi^*g_{WP}}, we see that the coefficients of {G} are linear combinations of the {\alpha}-pullbacks of {\partial a_j/\partial z_k} and {\partial a_j/\partial w_k}. Because {a_j} are (universally) bounded holomorphic functions, we can use Cauchy’s inequalities to see that the derivatives of {a_j} are (universally) bounded at any {(z,w)\in\Delta^N} with {\|(z,w)\|<1/2}. It follows from the boundedness of the (non-holomorphic) anti-diagonal embedding {\alpha} that the {k}-derivatives of {G_{ij}}‘s satisfy the desired bound. \Box

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 {C>0} and {\beta>0} such that, for any {X_0\in \mathcal{T}=Teich(S)}, the curvature tensor {R_{WP}} of the WP metric satisfies

\displaystyle \max\{\|\nabla R_{WP}(X_0)\|, \|\nabla^2 R_{WP}(X_0)\|\}\leq C d(X_0,\partial\mathcal{T})^{-\beta}

Proof: Fix {X_0\in Teich(S)} and consider the local coordinate {\psi=\psi_{X_0}} provided by Proposition 4. Since {\|D\psi\|} and {\|D\psi^{-1}\|} are uniformly bounded, our task is reduced to estimate the first two derivatives of the curvature tensor {R} of the metric {G(z)=\psi^*g_{WP}(z)=(G_{ij}(z))} at the origin {0\in \Delta^N}.

Recall that the Christoffel symbols of {G_{ij} = G_{ij}(z)} are

\displaystyle \Gamma^{m}_{ij}=\frac{1}{2}\sum\limits_{k}G^{mk}\left(\frac{\partial G_{ki}}{\partial \xi_j} + \frac{\partial G_{kj}}{\partial \xi_i} - \frac{\partial G_{ij}}{\partial \xi_k}\right)

or

\displaystyle \Gamma^{m}_{ij}=\frac{1}{2}G^{mk}(G_{ki,m} + G_{kj,i} - G_{ij,k})

in Einstein summation convention, and, in terms of the Christoffel symbols, the coefficients of the curvature tensor are

\displaystyle R^{l}_{ijk}=\frac{\partial\Gamma^{l}_{ik}}{\partial \xi_j} - \frac{\partial\Gamma^{l}_{ij}}{\partial \xi_k} + \Gamma^{l}_{js}\Gamma^{s}_{ik} - \Gamma^{l}_{ks}\Gamma^{s}_{ij}

Therefore, we see that the coefficients of the {k}-derivative {\nabla^k R} is a polynomial function of {G^{ij}} and the first {k+2} partial derivatives {G_{ij}} whose “degree” in the “variables” {G^{ij}} is {\leq k+2} (because of the formula {DG^{-1}(0)=-G^{-1}(0)\cdot DG(0)\cdot G^{-1}(0)}).

By Proposition 6, each {G^{ij}(0)} has order {O(\underline{\ell}(X_0)^{-2})} and the first {k+2} partial derivatives of {G_{ij}} at {0} are bounded by a constant depending only on {k}. It follows that

\displaystyle \begin{array}{rcl} \|\nabla^k R(0)\|^2&\leq& C(k)\sum\limits_{i_1,\dots,i_{k+3},j_1,\dots,j_{k+3},l,m}(\nabla^k R)^{l}_{i_1\dots i_{k+3}} (\nabla^k R)^{m}_{j_1\dots j_{k+3}} G^{i_1j_1}\dots G^{i_{k+3}j_{k+3}} G_{lm} \\ &\leq& C(k) \frac{1}{\underline{\ell}(X_0)^{2(k+2)}}\frac{1}{\underline{\ell}(X_0)^{2(k+2)}}\frac{1}{\underline{\ell}(X_0)^{2(k+3)}} = C(k)\frac{1}{\underline{\ell}(X_0)^{6k+14}}, \end{array}

and, consequently,

\displaystyle \max\{\|\nabla R_{WP}(X_0)\|,\|\nabla^2 R_{WP}(X_0)\|\}\leq C/\underline{\ell}(X_0)^{26/2}=C/d(X_0,\partial{T})^{26}.

This completes the proof. \Box

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 {R_{WP}} appearing in the proof of Theorem 7 are not sharp with respect to the exponent {\beta}. For instance, the WP metric on the moduli space {\mathcal{M}_{1,1}} of once-punctured torii has curvature {\sim -1/\ell\sim -1/d^2} where {d=d(X_0,\infty)} is the WP distance between {X_0} and the boundary {\partial \mathcal{M}_{1,1}=\{\infty\}}, so that one expects tha the {kth}-derivatives of the curvature behave like {\sim -1/d^{k+2}} (i.e., the exponent {6k+14} above should be {k+2}).In a very recent private communication, Wolpert indicated that it is possible to derive the sharp estimates of the form

\displaystyle \|\nabla^k R_{WP}(X_0)\|\leq C(k)/d(X_0,\partial\mathcal{T})^{k+2}

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 {c>0} such that for all {X\in\mathcal{M}[k]=\mathcal{T}/MCG[k]}, {k\geq 3}, one has the following polynomial lower bound on the injectivity radius of the WP metric at {X}:

\displaystyle inj(X)\geq c \cdot d_{WP}(X,\partial\mathcal{M}[k])^3

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 {c>0} such that, for any {\sigma\in\mathcal{C}(S)} and {X\in\mathcal{T}} with {\overline{\ell}(X)\ll 1},

\displaystyle d_{WP}(X,\Gamma(\sigma)(X))\geq c d(X,\partial\mathcal{T})^3

where {\Gamma(\sigma)\subset MCG(S)[k]} is the Abelian subgroup of the “level {k}” mapping class group {MCG(S)[k]} generated by the Dehn twists {\tau_{\alpha}} about the curves {\alpha\in\sigma}.

This reduces the proof of Theorem 8 to the following lemma:

Lemma 9 There exists an universal constant {J_0=J_0(g,n)\geq 1} with the following property. For each {\varepsilon>0}, there exists {\delta>0} such that, for any {X\in\mathcal{T}} with

\displaystyle d_{WP}(X,\phi(X))<\delta

for some non-trivial {\phi\in MCG(S)[k]}, one can find {\sigma\in\mathcal{C}(S)} so that {\overline{\ell}_{\sigma}(X)<\varepsilon} and {\phi^j\in\Gamma(\sigma)} for some {1\leq j\leq J_0}.

Proof: We begin the proof of the lemma by recalling that the mapping class group {MCG(S)[k]} acts on {\mathcal{T}} in a properly discontinuous way with no fixed points. Therefore, for each {\varepsilon>0}, there exists {\delta>0} such that if {d_{WP}(X,\phi(X))<\delta} for some non-trivial {\phi\in MCG(S)[k]} (i.e., some non-trivial element of the mapping class group has an “almost fixed point”), then {\overline{\ell}_{\sigma}(X)<\varepsilon} (i.e., the “almost fixed point” is close to the boundary of {\mathcal{T}}).

Let us show now that in the setting of the previous paragraph, {\phi^j\in \Gamma(\sigma)} for some {1\leq j\leq J_0}.

In this direction, let {J_0=J_0(g,n)\in\mathbb{N}} be the product of {(3g-3+n)!} 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 {X_m\in\mathcal{T}}, {\phi_m\in MCG(S)[k]}, {m\in\mathbb{N}}, such that {\overline{\ell}_{\sigma}(X_m)\ll 1} for some {\sigma\in\mathcal{C}(S)} and

\displaystyle \lim\limits_{m\rightarrow\infty}d(X_m,\phi_m(X_m))=0

but {\phi_m^j\notin\Gamma(\sigma)} for all {m\in\mathbb{N}}, {1\leq j\leq J_0}.

Passing to a subsequence (and applying appropriate elements of {\phi_m\in\Gamma(\sigma)}), we can assume that the sequence {X_m\in\mathcal{T}} converges to some noded Riemann surface {X_{\sigma}\in\partial\mathcal{T}_{\sigma}}. Because {d(X_m,\phi_m(X_m))\rightarrow 0} as {m\rightarrow\infty}, we see that ,for each {\beta\in\sigma},

\displaystyle \ell_{\phi_m(\beta)}(\phi_m(X))=\ell_{\beta}(X_m)\rightarrow 0.

It follows that, for all {m} sufficiently large, {\phi_m} sends any curve {\beta\in\sigma} to another curve {\phi_m(\beta)\in\sigma}. Therefore, for each {m} sufficiently large, there exists

\displaystyle 1\leq j=j(m) \leq \#\sigma!\leq (3g-3+n)!\leq J_0

such that {\phi_m^j} fixes each {\beta\in\sigma} (i.e., {\phi_m^j} is a reducible element of the mapping class group). By the Nielsen-Thruston classification of elements of the mapping class groups, the restrictions of {\phi_m^j} to each piece of {X_{\sigma}} 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 {S}).

It follows that we have only two possibilities for {\phi_m^j}: either the restriction of {\phi_m^j} to all pieces of {X_{\sigma}} are compositions of Dehn twists about certain curves in {\sigma} and finite order elements, or the restriction of {\phi_m^j} to some piece of {X_{\sigma}} is the composition of Dehn twists about certain curves in {\sigma} and a pseudo-Anosov element.

In the first scenario, by the definition of {J_0}, we can replace {\phi_m^j} by an adequate power {\phi_m^{J}} with {1\leq J\leq J_0} to “kill” the finite order elements and “keep” the Dehn twists. In other terms, {\phi_m^{J}\in \Gamma(\sigma)} (with {1\leq J\leq J_0}), a contradiction with our choice of the sequence {\phi_m}.

This leaves us with the second scenario. In this case, by definition of {J_0}, we can replace {\phi_m^j} by an adequate power {\phi_m^J} with {1\leq J\leq J_0} such that the restriction of {\phi_m^J} to some piece of {X_{\sigma}} is pseudo-Anosov. However, Daskalopoulos and Wentworth showed that there exists an uniform positive lower bound for

\displaystyle d_{WP}(X_{\sigma}, \phi_m^J(X_{\sigma}))

when {\phi_m^J} is pseudo-Anosov on some piece of {X_{\sigma}}. Since {1\leq J\leq J_0} and {J_0} is an universal constant, it follows that there exists an uniform positive lower bound for

\displaystyle d_{WP}(X_m,\phi_m(X_m))

for all {m} sufficiently large, a contradiction with our choice of the sequences {X_m\in\mathcal{T}} and {\phi_m\in MCG(S)[k]}.

These contradictions show that the sequences {X_m\in\mathcal{T}} and {\phi_m\in MCG(S)[k]} with the properties described above can’t exist.

This completes the proof of the lemma. \Box

3. First derivative of the Weil-Petersson flow

This section concerns the verification of item (VI) of Theorem 1 for the WP flow {\varphi_t}. More precisely, we will show the following result:

Theorem 10 There are constants {C\geq 1}, {\beta>0}, {\delta>0} and {\rho_0>0} such that

\displaystyle \|D_v\varphi_{\tau}\|_{WP}\leq C/\rho_{\tau}(v)^{\beta}

for any {0\leq\tau\leq\delta} and any {v\in T^1\mathcal{T}} with

\displaystyle 0<\rho_{\tau}(v):=\min\{d_{WP}(\varphi_t(v),\partial\mathcal{T}): t\in[-\tau,\tau]\}<\rho_0.

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 {\varphi_t} on arbitrary negatively curved manifold:

Theorem 11 Let {M} be a negatively curved manifold. Consider {\gamma:[-\tau,\tau]\rightarrow M} a geodesic where {0\leq \tau\leq 1} and suppose that for every {-\tau\leq t\leq\tau} the sectional curvatures of any plane containing {\dot{\gamma}(t)\in T^1M} is greater than {-\kappa(t)^2} for some Lipschitz function {\kappa:[-\tau,\tau]\rightarrow\mathbb{R}_+}.Then,

\displaystyle \|D_{\dot{\gamma}(0)}\varphi_{\tau}\|\leq 1+2(1+u(0)^2)(1+\sqrt{1+u(\tau)^2})\exp\left(\int_0^\tau u(s) ds\right)

where {u:[-\tau,\tau]\rightarrow[0,\infty)} is the solution of Riccati equation

\displaystyle u'+u^2=\kappa^2

with initial data {u(-\tau)=0}.

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 {\kappa(t)} for the sectional curvatures of the WP metric along WP geodesics {\gamma(t)}. More concretely, one has the following theorem:

Theorem 12 There are constants {Q, P, L\geq 2} and {0<\delta<1} such that for any {0<\delta'<\delta} and any geodesic segment {\gamma:(-\delta',\delta')\rightarrow\mathcal{T}} there exists a positive Lipschitz function {\kappa:(-\delta',\delta)\rightarrow \mathbb{R}_+} with

  • (a) {\sup\limits_{v\in T^1_{\gamma(t)}\mathcal{T}} -\langle R_{WP}(v,\dot{\gamma}(t)) \dot{\gamma}(t), v\rangle_{WP}\leq \kappa^2(t)} for all {t\in(-\delta',\delta')};
  • (b) {\kappa} is {Q}-controlled in the sense that {\kappa} has a right-derivative {D^+\kappa} satisfying

    \displaystyle D^+\kappa\geq \frac{1-Q^2}{Q}\kappa^2

  • (c) {\int_{-\delta'}^{\delta'} \kappa(s) ds\leq L|\log\rho_{\delta'}(\dot{\gamma}(0))|};
  • (d) {\max\{\kappa(0), \kappa(\delta')\}\leq P/\rho_{\delta'}(\dot{\gamma}(0))}.

Here, {\rho_{\delta'}(\dot{\gamma}(0))} denotes the distance between the geodesic segment {\gamma([- \delta', \delta])} and {\partial\mathcal{T}}.

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 {\kappa} the “WP curvature bound” function provided by Theorem 12 and let {u:[-\delta,\delta]\rightarrow\mathbb{R}_+} be the solution of Riccati’s equation

\displaystyle u'+u^2=\kappa^2

with initial data {u(-\delta)=0}.

Since {\kappa} is {Q}-controlled (in the sense of item (b) of Theorem 12), it follows that {u(t)\leq Q\kappa(t)} for all {t\in[-\delta,\delta]}: indeed, this is so because {u(-\delta)=0\leq Q\kappa(-\delta)}, and, if {u(t_0)=Q\kappa(t_0)} for some {t_0\in[-\delta,\delta]}, then

\displaystyle u'(t_0)=\kappa(t_0)^2-u(t_0)^2=(1-Q^2)\kappa(t_0)^2\leq Q\cdot D^+\kappa(t_0).

Therefore, by applying Theorem 11 in this setting, we deduce that

\displaystyle \|D_{\dot{\gamma}(0)}\varphi_{\tau}\|_{WP}\leq C/\rho_{\tau}(\dot{\gamma}(0))^{\beta}

for {\beta=L+3} and some constant {C=C(P,Q)\geq 1}. This completes the proof of Theorem 10. \Box

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 {\kappa} 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 {0} or {-\infty} only near the boundary, we can assume that our geodesic segment {\gamma:[-\delta',\delta']\rightarrow\mathcal{T}} in the statement of Theorem 12 is “relatively close” to a boundary stratum {\mathcal{T}_{\sigma}}, {\sigma\in\mathcal{C}(S)} (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 {\mathcal{T}}).

In this setting, for each {\alpha\in\sigma}, we consider the functions {f_{\alpha}(t):=\sqrt{\ell_{\alpha}(t)}} and

\displaystyle r_{\alpha}(t):=\sqrt{\langle \lambda_{\alpha}, \dot{\gamma}(t)\rangle^2 + \langle J\lambda_{\alpha}, \dot{\gamma}(t)\rangle^2}

(where {\lambda_{\alpha}:=\textrm{grad}\,\ell_{\alpha}^{1/2}}) along our geodesic segment {\gamma:I\rightarrow\mathcal{T}}, {I=(-\delta',\delta')}. 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 {\gamma(I)} are controlled in terms of {r_{\alpha}} and {f_{\alpha}}. Indeed, given {v\in T_{\gamma(t)}^1\mathcal{T}}, we can use a combined length basis {(\sigma,\chi)\in\mathcal{B}} to write

\displaystyle v:=\sum\limits_{\alpha\in\sigma}(a_{\alpha}\lambda_{\alpha}+b_{\alpha} J\lambda_{\alpha}) + \sum\limits_{\beta\in\chi} c_{\beta}\textrm{grad}\,\ell_{\beta}

Similarly, let us write

\displaystyle \dot{\gamma}(t)=\dot{\gamma}:=\sum\limits_{\alpha\in\sigma}(A_{\alpha}\lambda_{\alpha}+B_{\alpha} J\lambda_{\alpha}) + \sum\limits_{\beta\in\chi} C_{\beta}\textrm{grad}\,\ell_{\beta}

By Theorem 3, we obtain the following facts. Firstly, since {v} and {\dot{\gamma}} are WP-unit vectors, the coefficients {a_{\alpha}, b_{\alpha}, c_{\alpha}, A_{\alpha}, B_{\alpha}, C_{\alpha}} are

\displaystyle a_{\alpha}, b_{\alpha}, c_{\alpha}, A_{\alpha}, B_{\alpha}, C_{\alpha} = O(1)

Secondly, by definition of {r_{\alpha}}, we have that

\displaystyle r_{\alpha}^2=\frac{1}{4\pi^2}(A_{\alpha}^2+B_{\alpha}^2)+O(f_{\alpha}^3)

Finally,

\displaystyle \begin{array}{rcl} -\langle R_{WP}(v,\dot{\gamma})\dot{\gamma}, v\rangle_{WP} &=& \sum\limits_{\alpha\in\sigma} (a_{\alpha}^2B_{\alpha}^2+A_{\alpha}^2 b_{\alpha}^2) \langle R_{WP}(\lambda_{\alpha}, J\lambda_{\alpha})J\lambda_{\alpha}, \lambda_{\alpha}\rangle_{WP}+O(1) \\ &=&\sum\limits_{\alpha\in\sigma} O\left(\frac{r_{\alpha}^2}{f_{\alpha}^2}\right)+O(1) \end{array}

In summary, Wolpert’s formulas (Theorem 3) imply that

\displaystyle \sup\limits_{v\in T^1_{\dot{\gamma}(t)}\mathcal{T}} -\langle R_{WP}(v,\dot{\gamma}(t))\dot{\gamma}(t), v\rangle_{WP} = \sum\limits_{\alpha\in\sigma} O\left(\frac{r_{\alpha}(t)^2}{f_{\alpha}(t)^2}\right) \ \ \ \ \ (1)

(cf. Lemma 4.17 in Burns-Masur-Wilkinson paper).

Now, we want convert the expressions {r_{\alpha}(t)/f_{\alpha}(t)} into a positive Lipschitz function satisfying the properties described in items (b), (c), and (d) of Theorem 12, i.e., a {Q}-controlled function with appropriately bounded total integral and values at {0} and {\delta'}. 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

\displaystyle \kappa(t):=C\max_{\alpha\in\sigma}\left\{1,\frac{r_{\alpha}(t_{\alpha})}{r_{\alpha}(t_{\alpha})|t-t_{\alpha}|+f_{\alpha}(t_{\alpha})}\right\}

where {t_{\alpha}\in[-\delta',\delta']} is the (unique) time with {f_{\alpha}(t)\geq f_{\alpha}(t_{\alpha})} for all {t\in[-\delta',\delta']} and {C\geq 1} 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 {r_{\alpha}} and {f_{\alpha}}. For instance, the verification of item (a) (i.e., the fact that {\kappa} controls certain WP sectional curvatures along {\gamma}) relies on the fact that these two ODE’s permit to prove that

\displaystyle \frac{r_{\alpha}(t)}{f_{\alpha}(t)}\leq A\max\left\{1,\frac{r_{\alpha}(t_{\alpha})}{r_{\alpha}(t_{\alpha})|t-t_{\alpha}|+f_{\alpha}(t_{\alpha})}\right\}

for some sufficiently large constant {A\geq 1}. In particular, by plugging this into (1), we obtain that

\displaystyle \sup\limits_{v\in T^1_{\dot{\gamma}(t)}\mathcal{T}} -\langle R_{WP}(v,\dot{\gamma}(t))\dot{\gamma}(t), v\rangle_{WP}\leq \kappa^2(t),

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 {r_{\alpha}} and {f_{\alpha}}.

Lemma 13 (Lemma 4.15 of Burns-Masur-Wilkinson paper) {r_{\alpha}'(t)=O(f_{\alpha}^3(t))}.

Proof: By differentiating {r_{\alpha}(t)^2=\langle \lambda_{\alpha}, \dot{\gamma}(t)\rangle^2 + \langle J\lambda_{\alpha}, \dot{\gamma}(t)\rangle^2}, we see that

\displaystyle 2r_{\alpha}(t)r_{\alpha}'(t) = 2 \langle \lambda_{\alpha}, \dot{\gamma}(t)\rangle \langle \nabla_{\dot{\gamma}(t)}\lambda_{\alpha}, \dot{\gamma}(t)\rangle + 2 \langle J\lambda_{\alpha}, \dot{\gamma}(t)\rangle \langle J\nabla_{\dot{\gamma}(t)}\lambda_{\alpha}, \dot{\gamma}(t)\rangle.

Here, we used the fact that the WP metric is Kähler, so that {J} is parallel (“commutes with {\nabla}”).

Now, we observe that, by Wolpert’s formulas (cf. Theorem 3), one can write {\nabla_{\dot{\gamma}(t)}\lambda_{\alpha}} and {J\nabla_{\dot{\gamma}(t)}\lambda_{\alpha}} that

\displaystyle \nabla_{\dot{\gamma}(t)}\lambda_{\alpha}=\frac{3\langle\dot{\gamma}(t), J\lambda_{\alpha}\rangle}{2\pi f_{\alpha}(t)} J\lambda_{\alpha} + O(f_{\alpha}(t)^3)

and

\displaystyle J\nabla_{\dot{\gamma}(t)}\lambda_{\alpha}=-\frac{3\langle\dot{\gamma}(t), J\lambda_{\alpha}\rangle}{2\pi f_{\alpha}(t)} \lambda_{\alpha} + O(f_{\alpha}(t)^3)

Since {\max\{|\langle\dot{\gamma}(t), \lambda_{\alpha}|, |\langle\dot{\gamma}(t), J\lambda_{\alpha}\rangle|\}\leq r_{\alpha}(t)} (by definition), we conclude from the previous equations that

\displaystyle \begin{array}{rcl} 2r_{\alpha}(t)r_{\alpha}'(t) &=& \frac{3}{\pi f_{\alpha}(t)} (\langle \lambda_{\alpha}, \dot{\gamma}(t)\rangle \langle J\lambda_{\alpha}, \dot{\gamma}(t)\rangle^2 - \langle \lambda_{\alpha}, \dot{\gamma}(t)\rangle \langle J\lambda_{\alpha}, \dot{\gamma}(t)\rangle^2) \\ &+& O(r_{\alpha}(t) f_{\alpha}(t)^3) \\ &=& 0 + O(r_{\alpha}(t)f_{\alpha}(t)^3). \end{array}

This proves the lemma. \Box

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 \displaystyle y=x^3 .

Lemma 16 (Lemma 4.15 of Burns-Masur-Wilkinson paper)\displaystyle r_{\alpha}(t)^2 = f_{\alpha}'(t)^2 + \frac{2\pi}{3}f_{\alpha}(t) f_{\alpha}''(t)+O(f_{\alpha}(t)^4)

Proof: By definition, {\lambda_{\alpha}=\textrm{grad}\,\ell_{\alpha}^{1/2}}, so that

\displaystyle f_{\alpha}'(t)=\langle\lambda_{\alpha},\dot{\gamma}(t)\rangle.

Differentiating this equality and using Wolpert’s formulas (Theorem 3), we see that

\displaystyle f_{\alpha}''(t)=\langle\nabla_{\dot{\gamma}(t)}\lambda_{\alpha}, \dot{\gamma}(t)\rangle = \frac{3}{2\pi f_{\alpha}(t)}\langle\dot{\gamma}(t), J\lambda_{\alpha}\rangle^2 + O(f_{\alpha}(t)^3)

(Here, we used in the first equality the fact that {\gamma} is a geodesic, i.e., {\ddot{\gamma}(t)=0}.)

It follows that

\displaystyle \begin{array}{rcl} \frac{2\pi}{3}f_{\alpha}(t)f_{\alpha}''(t) + f_{\alpha}'(t)^2 &=& \langle\dot{\gamma}(t), J\lambda_{\alpha}\rangle^2 +\langle\dot{\gamma}(t), \lambda_{\alpha}\rangle^2 + O(f_{\alpha}(t)^4) \\ &=:& r_{\alpha}(t)^2+O(f_{\alpha}(t)^4). \end{array}

This proves the lemma. \Box


Responses

  1. Reblogged this on Weblog.


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Categories

%d bloggers like this: