Posted by: matheuscmss | September 6, 2019

Wolpert’s examples of tiny Weil-Petersson sectional curvatures revisited

In this previous post here (from 2018), I described some “back of the envelope calculations” (based on private conversations with Scott Wolpert) indicating that some sectional curvatures of the Weil–Petersson (WP) metric could be at least exponentially small in terms of the distance to the boundary divisor of Deligne–Mumford compactification.

Very roughly speaking, this heuristic computation went as follows: the WP sectional curvature of any {2}-plane can be written as the sum of three terms; for the {2}-planes considered in the previous post, the main term among those three seemed to be a kind of {L^4}-norm of Beltrami differentials with essentially disjoint supports; finally, this {L^4}-type norm was shown to be really small once a certain Green propagator is ignored.

Last April 2019, I met Scott during an event at Simons Center for Geometry and Physics, and I took the opportunity to tell him that one could perhaps show that the measure of the set of {2}-planes leading to tiny WP curvatures is very small using the real-analyticity of the WP metric.

More concretely, my idea was very simple: since the Grassmannian {G} of {2}-planes tangent to a point {p} is a compact space, the WP sectional curvature defines a real-analytic function {c:G\rightarrow(-\infty, 0)}, and we dispose of good upper bounds for {|c|} and all of its derivatives in terms of the distance of {p} to the boundary (see this article here), we can hope to get reasonable estimates for the measure of the sets {\{P\in G: |c(P)|\leq\varepsilon\}} using the techniques of these articles here and here (which are close in spirit to the classical fact [explained in Lemma 3.2 of Kleinbock–Margulis paper, for instance] that the measure of the sets {\{|P|\leq\varepsilon\}} are small whenever {P} is a polynomial function on {[0,1]} whose degree and {C^0}-norm are bounded).

As it turns out, Scott thought that this strategy made some sense and, in particular, he promised to use my suggestion as a motivation to review his arguments concerning WP sectional curvatures.

After several email exchanges with Howard Masur and I, Scott announced that there were some mistakes in the construction of tiny WP sectional curvature: in a nutshell, one should not restrict the analysis to a single “main term” in the formula for WP sectional curvatures as a sum of three expressions, and one can not ignore the effect of the Green propagator. More importantly, Scott made a detailed study of these mistakes which ultimately led him to establish polynomial upper bounds for WP sectional curvatures at the heart of his newest preprint available here.

In this post, we will follow closely Scott’s preprint in order to give an outline of the proof of a polynomial upper bound for WP sectional curvatures:

Theorem 1 (Wolpert) Given two integers {g\geq 0} and {n\geq 0} with {3g-3+n\geq 1}, there exists a constant {C(g,n)>0} with the following property.If {\sigma(X)} denotes the product of the lengths of the short geodesics of a hyperbolic surface {X} of genus {g} with {n} cusps whose systole is sufficiently small, then the sectional curvatures of the Weil-Petersson metric at {X} are at most

\displaystyle -C(g,n)\cdot\sigma(X)^7

Remark 1 As it was pointed out by Scott in his preprint, it is likely that this estimate is not optimal: indeed, one expects that the best exponent should be {3} rather than {7}.

In what follows, we’ll assume some familiarity with some basic aspects of the geometry of the Weil–Petersson metric (such as those described in these posts here and here).

1. Weil–Petersson sectional curvatures

Let {X} be a hyperbolic surface of genus {g\geq0} with {n\geq0}. If we write {X=\mathbb{H}/\Gamma}, where {\mathbb{H}} is the usual hyperbolic plane and {\Gamma} is a group of isometries of {\mathbb{H}} describing the fundamental group of {X}, then the holomorphic tangent space at {X} to the moduli space {\mathcal{M}_{g,n}} of Riemann surfaces of genus {g} with {n} punctures is naturally identified with the space {B(\Gamma)} of harmonic Beltrami differentials on {X} (and the cotangent space is related to quadratic differentials).

In this setting, the Weil–Petersson metric is the Riemannian metric {ds^2=2\sum g_{\alpha\overline{\beta}} dt_{\alpha}\overline{dt_{\beta}}} induced by the Hermitian inner product

\displaystyle g_{\alpha\overline{\beta}} = \langle\mu_{\alpha},\mu_{\beta}\rangle := \int_X \mu_{\alpha}\overline{\mu_{\beta}} \, dA

where {\mu_{\alpha}, \mu_{\beta}\in B(\Gamma)} and {dA} is the hyperbolic area form on {X}.

Remark 2 Note that {\langle.,.\rangle} is well-defined: if {\mu=\mu(z)\overline{dz}/dz} and {\nu=\nu(z)\overline{dz}/dz} are Beltrami differentials, then {\mu\overline{\nu}} is a function on {X}.

The Riemann tensor of the Weil–Petersson metric was computed by Wolpert in 1986:

\displaystyle R_{\alpha\overline{\beta}\gamma\overline{\delta}} = (\alpha\overline{\beta}, \gamma\overline{\delta}) + (\alpha\overline{\delta}, \gamma\overline{\beta})

where {(a\overline{b},c\overline{d}) := \int_X (\mu_a\overline{\mu_b}) \mathcal{D}(\mu_c\overline{\mu_d})\, dA} and {\mathcal{D}:=-2(\Delta-2)^{-1}} is an operator related to the Laplace–Beltrami operator {\Delta} on {L^2(X)}.

Remark 3 Our choice of notation here differs from Wolpert’s preprint! Indeed, he denotes the Laplace–Beltrami operator by {D} and he writes {\Delta=-2(D-2)^{-1}}.

The Riemann tensor gives access to nice formulas for the sectional curvatures thanks to the work of Bochner. More concretely, given {v_1} and {v_2} span a {2}-plane {P} in the real tangent space to {\mathcal{M}_{g,n}} at {X}, let us take Beltrami differentials {\mu_1} and {\mu_2} such that {v_1=\mu_1+\overline{\mu_2}}, {v_2=\mu_1-\overline{\mu_2}}, and {\{\mu_1,\mu_2\}} is orthonormal. Then, Bochner showed that the sectional curvature of {P} is

\displaystyle K(P)=\frac{R_{1\overline{2}1\overline{2}}-R_{1\overline{2}2\overline{1}}-R_{2\overline{1}1\overline{2}}+R_{2\overline{1}2\overline{1}}}{4g_{1\overline{1}}g_{2\overline{2}}-2|g_{1\overline{2}}|^2-2\textrm{Re}(g_{1\overline{2}})^2} = \frac{R_{1\overline{2}1\overline{2}}-R_{1\overline{2}2\overline{1}}-R_{2\overline{1}1\overline{2}}+R_{2\overline{1}2\overline{1}}}{4}

Hence, by Wolpert’s formula for the Riemann tensor of the WP metric, we see that

\displaystyle K(P) = \frac{2(1\overline{2}, 1\overline{2})-(1\overline{2}, 2\overline{1})-(1\overline{1}, 2\overline{2})-(2\overline{1}, 1\overline{2})-(2\overline{2}, 1\overline{1})+2(2\overline{1}, 2\overline{1})}{4} \ \ \ \ \ (1)

2. Spectral theory of {\mathcal{D}}

Wolpert’s formula for the Riemann tensor of the WP metric hints that the spectral theory of {\mathcal{D}} plays an important role in the study of the WP sectional curvatures.

For this reason, let us review some key properties of {\mathcal{D}} (and we refer to Section 3 of Wolpert’s preprint for more details and references). First, {\mathcal{D}=-2(\Delta-2)^{-1}} is a positive operator on {L^2(X)} whose norm is {1}: these facts follow by integration by parts. Secondly, {\Delta} is essentially self-adjoint on {L^2(X)}, so that {\mathcal{D}} is self-adjoint on {L^2(X)}. Moreover, the maximum principle permits to show that {\mathcal{D}} is also a positive operator on {C_0(X)} with unit norm. Finally, {\mathcal{D}} has a positive symmetric integral kernel: indeed,

\displaystyle \mathcal{D}f(p) = \int_X G(p,q) f(q) \, dA

where the Green propagator {G} is the Poincaré series

\displaystyle G(p,q)=-2\sum\limits_{\gamma\in\Gamma} Q_1(d(p,\gamma(q)))

associated to an appropriate Legendre function {Q_1}. (Here, {d(.,..)} stands for the hyperbolic distance on {\mathbb{H}}.) For later reference, we recall that {Q_1} has a logarithmic singularity at {0} and {-Q_1(x)\sim e^{-2x}} whenever {x} is large.

3. Negativity of the WP sectional curvatures

Interestingly enough, as it was first noticed by Wolpert in 1986, the spectral features of {\mathcal{D}} described above are sufficient to derive the negativity of WP sectional curvatures from Cauchy-Schwarz inequality. More precisely, since {\mathcal{D}} is self-adjoint, i.e.,

\displaystyle (a\overline{b},c\overline{d}) := \int_X (\mu_a\overline{\mu_b}) \, \mathcal{D}(\mu_c\overline{\mu_d}) \, dA = \int_X \mathcal{D}(\mu_a\overline{\mu_b}) \, \mu_c\overline{\mu_d}\,dA

and its integral kernel {G} is a real function, a straightforward computation reveals that the equation (1) for the sectional curvature {K(P)} of a {2}-plane {P} can be rewritten as

\displaystyle \begin{array}{rcl} K(P) &=& \frac{2(1\overline{2}, 1\overline{2})-(1\overline{2}, 2\overline{1})-(1\overline{1}, 2\overline{2})-(2\overline{1}, 1\overline{2})-(2\overline{2}, 1\overline{1})+2(2\overline{1}, 2\overline{1})}{4} \\ &=& \frac{4\textrm{Re}(1\overline{2}, 1\overline{2}) -2(1\overline{2}, 2\overline{1}) -2(1\overline{1}, 2\overline{2})}{4}. \end{array}

If we decompose the function {\mu_1\overline{\mu_2}} into its real and imaginary parts, say {\mu_1\overline{\mu_2} = f+ih}, then we see that

\displaystyle \begin{array}{rcl} \textrm{Re}(1\overline{2}, 1\overline{2}) - (1\overline{2}, 2\overline{1}) &=& \left[\int_X f\,\mathcal{D}f \, dA - \int_X h\, \mathcal{D}h \, dA\right] - \left[\int_X f\,\mathcal{D}f \, dA + \int_X h\, \mathcal{D}h \, dA\right] \\ &=& -2\int_X h\, \mathcal{D}h \, dA. \end{array}

Since {\mathcal{D}} is a positive operator, we conclude that {\textrm{Re}(1\overline{2}, 1\overline{2}) - (1\overline{2}, 2\overline{1})\leq 0} and, a fortiori,

\displaystyle K(P)\leq \frac{\textrm{Re}(1\overline{2}, 1\overline{2})-(1\overline{1}, 2\overline{2})}{2} \ \ \ \ \ (2)

The non-positivity of the right-hand side of (2) can be established in three steps. First, the positivity of {\mathcal{D}} also implies that

\displaystyle \textrm{Re}(1\overline{2}, 1\overline{2})\leq \int_X |f|\,\mathcal{D}|f|\,dA\leq \int_X |f|\,\mathcal{D}|\mu_1\overline{\mu_2}|\,dA.

Secondly, the fact that {\mathcal{D}} has a positive integral kernel {G} allows to apply the Cauchy–Schwarz inequality to get that {\mathcal{D}|uv| =\int G |u v| = \int G^{1/2}|u| G^{1/2}|v| \leq (\mathcal{D}|u|^2)^{1/2} (\mathcal{D}|v|^2)^{1/2}}. Therefore,

\displaystyle \int_X |f|\,\mathcal{D}|\mu_1\overline{\mu_2}|\,dA\leq \int_X |f|\,(\mathcal{D}|\mu_1|^2)^{1/2} (\mathcal{D}|\mu_2|^2)^{1/2}\,dA\leq \int_X |\mu_1\overline{\mu_2}| \, (\mathcal{D}|\mu_1|^2)^{1/2} (\mathcal{D}|\mu_2|^2)^{1/2}\,dA

Finally, the Cauchy–Schwarz inequality also says that

\displaystyle \int_X |\mu_1\overline{\mu_2}| \, (\mathcal{D}|\mu_1|^2)^{1/2} (\mathcal{D}|\mu_2|^2)^{1/2}\,dA\leq \left(\int_X |\mu_1|^2\,(\mathcal{D}|\mu_2|^2)\, dA\right)^{1/2}\left(\int_X |\mu_2|^2\,(\mathcal{D}|\mu_1|^2)\, dA\right)^{1/2}=(1\overline{1},2\overline{2})

In summary, we showed that

\displaystyle (I)\leq (II)\leq (III)\leq (IV)\leq (V)\leq (VI) \ \ \ \ \ (3)

where

\displaystyle \begin{array}{rcl} & &(I):=\textrm{Re}(1\overline{2}, 1\overline{2}), \quad (II):=\int_X |f|\,\mathcal{D}|f|\,dA, \quad (III):=\int_X |f|\,\mathcal{D}|\mu_1\overline{\mu_2}|\,dA, \\ & & (IV):=\int_X |f|\,(\mathcal{D}|\mu_1|^2)^{1/2} (\mathcal{D}|\mu_2|^2)^{1/2}\,dA, \quad (V):=\int_X |\mu_1\overline{\mu_2}| \, (\mathcal{D}|\mu_1|^2)^{1/2} (\mathcal{D}|\mu_2|^2)^{1/2}\,dA, \\ & & (VI):= (1\overline{1}, 2\overline{2}) \end{array}

In particular, {(I)\leq (VI)}, so that it follows from (2) that all sectional curvatures {K(P)} of the WP metric are non-positive, i.e., {K(P)\leq 0}.

Actually, it is not hard to derive that {K(P)<0} at this stage: indeed, {K(P)=0} would force a case of equality in Cauchy-Schwarz inequality and this is not possible in our context because {\{\mu_1,\mu_2\}} is orthonormal.

Remark 4 Philosophically speaking, the “analog” to this argument in the realm of Teichmüller dynamics is Forni’s proof of the spectral gap property {\lambda_2<1} for the Lyapunov exponents of the Teichmüller geodesic flow. In fact, after some computations with variational formulas for the so-called Hodge norm, Forni establishes that {\lambda_2<1} by ruling out an equality case in a certain Cauchy-Schwarz estimate.

4. Reduction of Theorem 1 to bounds on {\mathcal{D}}‘s kernel

The discussion in the previous section says that small WP sectional curvatures correspond to almost equalities in certain Cauchy-Schwarz inequalities.

Hence, a natural strategy towards the proof of Theorem 1 consists into showing that an almost equality in (3) is impossible. In this direction, Wolpert establishes the following result:

Theorem 2 (Wolpert) There are two constants {c_1(g,n)>0} and {c_2(g,n)>0} with the following property. If we have an almost equality

\displaystyle (V)-(I)\leq c_1(g,n)\cdot\sigma(X)^7,

between the terms {(I)} and {(V)} in (3), then {(VI)} and {(I)} can not be almost equal:

\displaystyle (VI)-(I)\geq c_2(g,n)\cdot\sigma(X)^3

Of course, Theorem 1 is an immediate consequence of Theorem 2 (in view of (2) and the estimate {(VI)-(I)\geq (V)-(I)} [implied by (3)]).

Thus, it remains only to prove Theorem 2. For this sake, we need further spectral information on {\mathcal{D}}, namely, some lower bounds on its the kernel {G(p,q)}. In order to illustrate this point, let us now show Theorem 2 assuming the following statement.

Proposition 3 There exists a constant {c_3(g,n)>0} such that

\displaystyle G(p,q)\geq c_3(g,n)\cdot \sigma(X)^3

whenever {p} and {q} do not belong to the cusp region {X_{cusps}} of {X}.

Remark 5 We recall that the cusp region {X_{cusps}} of {X} is a finite union of portions of {X} which are isometric to a punctured disk {\{0<|w|<c_4(g,n)\}} (equipped with the hyperbolic metric {ds^2=(|dw|/|w|\log|w|)^2}).

For the sake of exposition, let us first establish Theorem 2 when {X} is compact, i.e., {X_{cusps}=\emptyset}, before explaining the extra ingredient needed to treat the general case.

4.1. Proof of Theorem 2 modulo Proposition 3 when {X_{cusps}=\emptyset}

Suppose that {(V)-(I)\leq c_1(g,n)\sigma(X)^7} for a constant {c_1(g,n)} to be chosen later. In this regime, our goal is to show that {(VI)} is “big” and {(II)} is “small”, so that {(VI)-(I)} is necessarily “big”.

We start by quickly showing that {(VI)} is “big”. Since {\mu_1} and {\mu_2} are unitary tangent vectors, it follows from Proposition 3 that

\displaystyle (VI)=\int_X |\mu_1|^2\,\mathcal{D}|\mu_2|^2\,dA\geq c_3(g,n) \sigma(X)^3 \ \ \ \ \ (4)

Let us now focus on proving that {(II)} is “small”. Since {(II)-(I)\leq (V)-(I)} (cf. (3)), if we write {\mu_1\overline{\mu_2} = f+ih=f^+-f^-+ih} (where {f^+} and {f^-} are the positive and negative parts of the real part {f} of {\mu_1\overline{\mu_2})}, then we obtain that

\displaystyle \begin{array}{rcl} c_1(g,n)\,\sigma(X)^7\geq (II)-(I) &=& \int_X |f|\,\mathcal{D}|f|\,dA - \textrm{Re}\int_X\mu_1\overline{\mu_2}\,\mathcal{D}(\mu_1\overline{\mu_2})\, dA \\ &=& \int_X f^+\,\mathcal{D}f^+\,dA + 2\int_X f^+\,\mathcal{D}f^-\,dA+\int_X f^-\,\mathcal{D}f^-\,dA \\ & &- \int_X f\,\mathcal{D}f\,dA+\int_X h\,\mathcal{D}h\,dA \\ &=&4\int_X f^+\,\mathcal{D}f^-\,dA+\int_X h\,\mathcal{D}h\,dA. \end{array}

Since {\mathcal{D}} is positive, we derive that {4\int_X f^+\,\mathcal{D}f^-\,dA\leq c_1(g,n)\,\sigma(X)^7}. Thus, if {X} is compact, i.e., {X_{cusps}=\emptyset}, then Proposition 3 says that {G(p,q)\geq c_3(g,n)\,\sigma(X)^3} for all {p,q\in X}. It follows that

\displaystyle 4\,c_3(g,n)\,\sigma(X)^3\int_X f^+\,dA\int_X f^-\,dA\leq c_1(g,n)\,\sigma(X)^7

By orthogonality of {\{\mu_1,\mu_2\}}, we have that {\textrm{Re}\int_X\mu_1\overline{\mu_2}\,dA=0}, i.e., {\int_X f^+\,dA = \int_X f^-\,dA = (1/2) \int_X |f|\,dA}. By plugging this information into the previous inequality, we obtain the estimate

\displaystyle c_3(g,n)\,\left(\int_X |f|\,dA\right)^2\leq c_1(g,n)\,\sigma(X)^4 \ \ \ \ \ (5)

Next, we observe that {(V)-(IV)\leq (V)-(I)} (cf. (3)) in order to obtain that

\displaystyle c_1(g,n)\,\sigma(X)^7\geq (V)-(IV)=\int_X (|\mu_1\overline{\mu_2}|-|f|) \, (\mathcal{D}|\mu_1|^2)^{1/2} (\mathcal{D}|\mu_2|^2)^{1/2}\,dA

On the other hand, Proposition 3 ensures that {\mathcal{D}|\mu_{\ast}|^2\geq c_3(g,n)\,\sigma(X)^3\,\int_X|\mu_{\ast}|^2\,dA} for {\ast=1,2}. Since {\mu_1} and {\mu_2} are unitary tangent vectors, one has {\mathcal{D}|\mu_{\ast}|^2\geq c_3(g,n)\,\sigma(X)^3} for {\ast=1,2}. By inserting this inequality into the previous estimate, we derive that

\displaystyle c_1(g,n)\,\sigma(X)^7\geq c_3(g,n)\,\sigma(X)^3\,\int_X (|\mu_1\overline{\mu_2}|-|f|)\,dA \ \ \ \ \ (6)

From (5) and (6), we see that

\displaystyle \int_X |\mu_1\overline{\mu_2}|\,dA\leq \sqrt{\frac{c_1(g,n)}{c_3(g,n)}}\sigma(X)^2+\frac{c_1(g,n)}{c_3(g,n)}\sigma(X)^4\leq 2\sqrt{\frac{c_1(g,n)}{c_3(g,n)}}\sigma(X)^2 \ \ \ \ \ (7)

whenever {X} has a sufficiently small systole.

This {L^1} bound on {|\mu_1\overline{\mu_2}|} can be converted into a {C^0} bound thanks to Cauchy integral formula. More concrentely, as it is explained in Section 2 of Wolpert’s preprint, after observing that {|\mu_1\overline{\mu_2}| = |\mu_1\mu_2|} and replacing Beltrami differentials {\mu_1} and {\mu_2} by the dual objects {q_1} and {q_2} (namely, quadratic differentials), we are led to study quartic differentials {q_1q_2}. By Cauchy integral formula on {\mathbb{H}}, one has

\displaystyle |q_1q_2(ds^2)^{-2}|(p)\leq \frac{1}{\pi}\int_{B(p,1)}|q_1q_2(ds^2)^{-2}|\,dA

On the other hand, if {X=\mathbb{H}/\Gamma} has systole {\rho(X)} and the cusp region {X_{cusps}} is empty, then the injectivity radius at any {p\in X} is {\geq \rho(X)/2}. Thus, there exists an universal constant {c_0>0} such that

\displaystyle |q_1q_2(ds^2)^{-2}|(p)\leq \frac{1}{\pi}\int_{B(p,1)}|q_1q_2(ds^2)^{-2}|\,dA\leq c_0\frac{1}{\rho(X)}\|q_1q_2\|_{L^1(X)}

for all {p\in X}. By plugging this inequality into (7), we conclude that

\displaystyle |\mu_1\overline{\mu_2}(p)|\leq 2c_0\sqrt{\frac{c_1(g,n)}{c_3(g,n)}}\frac{1}{\rho(X)}\sigma(X)^2\leq 2c_0\sqrt{\frac{c_1(g,n)}{c_3(g,n)}}\sigma(X)

for all {p\in X}.

Since {\mathcal{D}} is a positive operator on {C_0(X)} with unit norm (cf. Section 2 above) and {|f|\leq |\mu_1\overline{\mu_2}|}, we have that the previous inequality implies the following {C^0} bound on {\mathcal{D}|f|}:

\displaystyle \mathcal{D}|f|(p)\leq 2c_0\sqrt{\frac{c_1(g,n)}{c_3(g,n)}}\sigma(X)

for all {p\in X}. By combining this estimate with (7), we conclude that

\displaystyle (II)=\int_X |f|\,\mathcal{D}|f|\,dA\leq 4c_0\frac{c_1(g,n)}{c_3(g,n)}\sigma(X)^3 \ \ \ \ \ (8)

In summary, (4) and (8) imply that

\displaystyle (VI)-(I)\geq (VI)-(II)\geq \frac{c_3(g,n)}{2}\cdot\sigma(X)^3:=c_2(g,n)\cdot\sigma(X)^3

for the choice of constant {c_1(g,n):=\frac{c_3(g,n)^2}{8c_0}}. This proves Theorem 2 in the absence of cusp regions.

4.2. Proof of Theorem 2 modulo Proposition 3 when {X_{cusps}\neq\emptyset}

The arguments above for the case {X_{cusps}=\emptyset} also work in the case {X_{cusps}\neq\emptyset} because the cusp regions carry only a tiny fraction of the mass of the relevant functions, Beltrami differentials, etc.

More precisely, as it is explained in Section 2 of Wolpert’s preprint, if the constant {c_4(g,n)>0} is chosen correctly, then the Cauchy integral formula and the Schwarz lemma can be used to prove that

\displaystyle \int_{X_{cusps}}|\varphi (ds^2)^{-2}|\,dA\leq \frac{1}{8}\|\varphi\|_{L^1(X)}

for all holomorphic quartic differentials {\varphi}.

In particular, we do not lose too much information after truncating {\mu_1}, {\mu_2}, etc. to {X\setminus X_{cusps}} and this allows us to repeat the arguments of the case {X_{cusps}=\emptyset} to the corresponding truncated objects {\widetilde{\mu_1}}, {\widetilde{\mu_2}}, etc. without any extra difficulty: see Section 5 of Wolpert’s preprint for more details.

5. Proof of Proposition 3

Closing this post, let us give an idea of the proof of Proposition 3 (and we refer the reader to Section 4 of Wolpert’s preprint for more details).

Since {G(p,q)=-2\sum\limits_{\gamma\in\Gamma} Q_1(d(p,\gamma(q)))} and {-Q_1\sim e^{-2x}} (cf. Section 2 above), our task is reduced to give lower bounds on the Poincaré series

\displaystyle K(p,q)=\sum\limits_{\gamma\in\Gamma} e^{-2d(p,\gamma(q))}

For this sake, let us first recall that a hyperbolic surface {X} has thick-thin decomposition: the thick portion is the region where the injectivity radius is bounded away from zero by a uniform constant and the thin portion is the complement of the thick region. Geometrically, the thin region is the disjoint union of the cusp region {X_{cusps}} and a finite number of collars around simple closed short geodesics: roughly speaking, a collar consisting of the points at distance {\leq w(\alpha)=\log(1/\ell_{\alpha})+O(1)} of a short simple closed geodesic {\alpha} of length {\ell_{\alpha}}.

We can provide lower bounds on {K(p,q)} in terms of the behaviours of simple geodesic arcs connecting {p} and {q} on {X}.

More concretely, let {\theta_{pq}} be the shortest geodesic connecting {p} and {q}. Since {\theta_{pq}} is simple, we have that, for certain adequate choices of the constants defining the collars, one has that {\theta_{pq}} can not “back track” after entering a collar, i.e., it must connect the boundaries (rather than going out via the same boundary component). Furthermore, {\theta_{pq}} can not go very high into a cusp. Thus, if we decompose {\theta_{pq}} according to its visits to the thick region, the collars and the cusps, then the fact that {p,q\in X\setminus X_{cusps}} permits to check that it suffices to study the passages of {\theta_{pq}} through collars in order to get a lower bound on {K(p,q)}.

Next, if {\eta} is a subarc of {\theta_{pq}} crossing a collar around a short closed geodesic {\alpha}, then we can apply Dehn twists to {\eta} to get a family of simple arcs indexed by {\mathbb{Z}} giving a “contribution” to {K(p,q)} of

\displaystyle \sum\limits_{n\in\mathbb{Z}}e^{-2(2w(\alpha)+|n|\ell_{\alpha})}\geq c_5(g,n)\cdot \ell_{\alpha}^3

for some constant {c_5(g,n)>0} depending only on the topology of {X}. In this way, the desired result follows by putting all “contributions” together.


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