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 -plane can be written as the sum of three terms; for the -planes considered in the previous post, the main term among those three *seemed* to be a kind of -norm of Beltrami differentials with essentially disjoint supports; finally, this -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 -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 of -planes tangent to a point is a compact space, the WP sectional curvature defines a real-analytic function , and we dispose of good upper bounds for and all of its derivatives in terms of the distance of to the boundary (see this article here), we can hope to get reasonable estimates for the measure of the sets 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 are small whenever is a polynomial function on whose degree and -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 and with , there exists a constant with the following property.If denotes the product of the lengths of the short geodesics of a hyperbolic surface of genus with cusps whose systole is sufficiently small, then the sectional curvatures of the Weil-Petersson metric at are at most

Remark 1As 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 rather than .

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 be a hyperbolic surface of genus with . If we write , where is the usual hyperbolic plane and is a group of isometries of describing the fundamental group of , then the holomorphic *tangent* space at to the moduli space of Riemann surfaces of genus with punctures is naturally identified with the space of harmonic Beltrami differentials on (and the *cotangent* space is related to quadratic differentials).

In this setting, the Weil–Petersson metric is the Riemannian metric induced by the Hermitian inner product

where and is the hyperbolic area form on .

Remark 2Note that is well-defined: if and are Beltrami differentials, then is a function on .

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

where and is an operator related to the Laplace–Beltrami operator on .

Remark 3Our choice of notation here differs from Wolpert’s preprint! Indeed, he denotes the Laplace–Beltrami operator by and he writes .

The Riemann tensor gives access to nice formulas for the sectional curvatures thanks to the work of Bochner. More concretely, given and span a -plane in the real tangent space to at , let us take Beltrami differentials and such that , , and is orthonormal. Then, Bochner showed that the sectional curvature of is

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

**2. Spectral theory of **

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

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

where the Green propagator is the Poincaré series

associated to an appropriate Legendre function . (Here, stands for the hyperbolic distance on .) For later reference, we recall that has a logarithmic singularity at and whenever is large.

**3. Negativity of the WP sectional curvatures**

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

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

If we decompose the function into its real and imaginary parts, say , then we see that

Since is a positive operator, we conclude that and, *a fortiori*,

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

Secondly, the fact that has a positive integral kernel allows to apply the Cauchy–Schwarz inequality to get that . Therefore,

Finally, the Cauchy–Schwarz inequality also says that

In particular, , so that it follows from (2) that all sectional curvatures of the WP metric are non-positive, i.e., .

Actually, it is not hard to derive that at this stage: indeed, would force a case of equality in Cauchy-Schwarz inequality and this is not possible in our context because is orthonormal.

Remark 4Philosophically speaking, the “analog” to this argument in the realm of Teichmüller dynamics is Forni’s proof of the spectral gap property 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 by ruling out an equality case in a certain Cauchy-Schwarz estimate.

**4. Reduction of Theorem 1 to bounds on ‘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 and with the following property. If we have an almost equality

between the terms and in (3), then and can not be almost equal:

Of course, Theorem 1 is an immediate consequence of Theorem 2 (in view of (2) and the estimate [implied by (3)]).

Thus, it remains only to prove Theorem 2. For this sake, we need further spectral information on , namely, some lower bounds on its the kernel . In order to illustrate this point, let us now show Theorem 2 *assuming* the following statement.

Proposition 3There exists a constant such that

whenever and do not belong to the cusp region of .

Remark 5We recall that the cusp region of is a finite union of portions of which are isometric to a punctured disk (equipped with the hyperbolic metric ).

For the sake of exposition, let us first establish Theorem 2 when is compact, i.e., , before explaining the extra ingredient needed to treat the general case.

**4.1. Proof of Theorem 2 modulo Proposition 3 when **

Suppose that for a constant to be chosen later. In this regime, our goal is to show that is “big” and is “small”, so that is necessarily “big”.

We start by quickly showing that is “big”. Since and are unitary tangent vectors, it follows from Proposition 3 that

Let us now focus on proving that is “small”. Since (cf. (3)), if we write (where and are the positive and negative parts of the real part of , then we obtain that

Since is positive, we derive that . Thus, if is compact, i.e., , then Proposition 3 says that for all . It follows that

By orthogonality of , we have that , i.e., . By plugging this information into the previous inequality, we obtain the estimate

Next, we observe that (cf. (3)) in order to obtain that

On the other hand, Proposition 3 ensures that for . Since and are unitary tangent vectors, one has for . By inserting this inequality into the previous estimate, we derive that

whenever has a sufficiently small systole.

This bound on can be converted into a bound thanks to Cauchy integral formula. More concrentely, as it is explained in Section 2 of Wolpert’s preprint, after observing that and replacing Beltrami differentials and by the dual objects and (namely, quadratic differentials), we are led to study quartic differentials . By Cauchy integral formula on , one has

On the other hand, if has systole and the cusp region is empty, then the injectivity radius at any is . Thus, there exists an universal constant such that

for all . By plugging this inequality into (7), we conclude that

for all .

Since is a positive operator on with unit norm (cf. Section 2 above) and , we have that the previous inequality implies the following bound on :

for all . By combining this estimate with (7), we conclude that

In summary, (4) and (8) imply that

for the choice of constant . This proves Theorem 2 in the absence of cusp regions.

**4.2. Proof of Theorem 2 modulo Proposition 3 when **

The arguments above for the case also work in the case 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 is chosen correctly, then the Cauchy integral formula and the Schwarz lemma can be used to prove that

for all holomorphic quartic differentials .

In particular, we do not lose too much information after truncating , , etc. to and this allows us to repeat the arguments of the case to the corresponding truncated objects , , 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 and (cf. Section 2 above), our task is reduced to give lower bounds on the Poincaré series

For this sake, let us first recall that a hyperbolic surface 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 and a finite number of *collars* around simple closed short geodesics: roughly speaking, a collar consisting of the points at distance of a short simple closed geodesic of length .

We can provide lower bounds on in terms of the behaviours of simple geodesic arcs connecting and on .

More concretely, let be the shortest geodesic connecting and . Since is simple, we have that, for certain adequate choices of the constants defining the collars, one has that 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, can not go very high into a cusp. Thus, if we decompose according to its visits to the thick region, the collars and the cusps, then the fact that permits to check that it suffices to study the passages of through collars in order to get a lower bound on .

Next, if is a subarc of crossing a collar around a short closed geodesic , then we can apply Dehn twists to to get a family of simple arcs indexed by giving a “contribution” to of

for some constant depending only on the topology of . In this way, the desired result follows by putting all “contributions” together.

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