During the preparation of my joint articles with K. Burns, H. Masur and A. Wilkinson about the rates of mixing of the Weil-Peterson geodesic flow (on moduli spaces of Riemann surfaces), we exchanged some emails with S. Wolpert about the sectional curvatures of the Weil-Petersson metric near the boundary of moduli spaces.

As it turns out, Wolpert communicated to us an interesting mechanism to show that some sectional curvatures can be *exponentially* small in terms of the square of the distance to the boundary.

On the other hand, this mechanism does not seem to be well-known: indeed, I was asked in many occasions about the behavior of the Weil-Petersson sectional curvatures near the boundary, and each time my colleagues were surprised by Wolpert’s examples.

In this short post, I will try to describe Wolpert’s construction of tiny Weil-Petersson sectional curvatures. (Of course, all mistakes below are my responsibility.)

**1. Weil-Petersson metric**

Recall that the cotangent bundle to the moduli space of Riemann surfaces is naturally identified with the space of quadratic differentials on Riemann surfaces.

The Weil-Petersson inner product between two quadratic differentials and on a Riemann surface is

where is the hyperbolic metric of .

Remark 1The quadratic differentials and are locally given by and , so that . In particular, we use the hyperbolic metric to obtain a -type formula (because the area form is ).

The incomplete, smooth, Kähler, negatively curved Riemannian metrics on moduli spaces of Riemann surfaces induce by the Weil-Petersson inner products are the so-called Weil-Petersson (WP) metrics.

Recall that the moduli spaces of Riemann surfaces are not compact because a hyperbolic closed geodesic on a Riemann surface might have arbitrarily small hyperbolic length . Moreover, the Weil-Petersson metric is incomplete because we can pinch off a hyperbolic closed geodesic on in finite time . Nevertheless, the natural compactification of the moduli space of Riemann surfaces with respect to the Weil-Petersson metric turns out to be the Deligne-Mumford compactification where one adds a boundary by including stable nodal Riemann surfaces into the picture. (See Burns-Masur-Wilkinson paper and the references therein for more details.)

Today, we are interested on the order of magnitude of the Weil-Petersson sectional curvatures at a point of moduli space of Riemann surfaces. More concretely, we want to understand WP sectional curvatures of cotangent planes to in terms of the distance of to the boundary (of Deligne-Mumford compactification).

By Wolpert’s work, we know that , i.e., WP sectional curvatures are bounded away from by a *polynomial* function of the inverse of the distance to the boundary.

On the other hand, a potential *cancellation* in Wolpert’s formulas for WP curvatures makes it hard to infer upper bounds on WP sectional curvatures in terms of . (Nevertheless, the situation is better understood for *holomorphic* sectional curvatures and WP Ricci curvatures: see, e.g., Melrose-Zhu paper.)

In any event, Wolpert discovered that there is no chance to expect an upper bound on all WP sectional curvatures at in terms of a polynomial function of the distance of to the boundary: in fact, we will see below that Wolpert constructed examples of Riemann surfaces where some WP sectional curvature behaves like .

**2. Plumbing coordinates**

The geometry of a Riemann surface near the boundary of moduli space is described by the so-called *plumbing construction*.

Roughly speaking, if a Riemann surface is close to acquire a node at a curve , then we can describe an annular region surrounding using a complex parameter with and two complex coordinates and with the following properties.

The curve separates the annular region into two components and . The coordinate takes to in such a way is mapped to and is mapped into . Similarly, the coordinate takes to in such a way is mapped to and is mapped into . Furthermore, we recover the annular region by identifying points via the relation

In the figure below, we depicted a Riemann surface obtained from this plumbing construction near a curve separating it into two torii.

Remark 2In the plumbing construction, the size of the parameter gives a bound on the distance of to the boundary of moduli space: indeed, the hyperbolic length of the geodesic representative of is . Of course, this is coherent with the idea that describes a node.Also, the phase of is related to the so-called twist parameters.

**3. Tiny Weil-Petersson curvature**

Consider the plumbing construction in the figure above. It illustrates a curve separating a Riemann surface of genus into two torii and with natural coordinates and . In these coordinates, the curve is .

We start with the quadratic differential on . If we want to extend to and, *a fortiori*, , then we need to understand the behavior of in the portion of intersecting the annular region surrounding . In other words, we have to describe in -coordinates.

For this sake, we recall that the definition of plumbing construction says that . Thus, and the formula allows us to extend to .

This description has the following interesting consequence: while the Weil-Petersson size of on is (because ), the Weil-Petersson size (cf. (1)) of on is because

By exchanging the roles of the subindices and in the previous discussion, we also get a quadratic differential with Weil-Petersson size on and on .

Remark 3The reader is invited to consult Sections 2 and 3 of this paper of Wolpert for a more detailed discussion of quadratic differentials on Riemann surfaces coming from plumbing constructions.

At this point, Wolpert notices that the Weil-Petersson sectional curvature of the plane spanned by and is tiny in the following sense.

It is explained in this paper of Wolpert that the Weil-Petersson curvature is a sort of “-norm” which in the case of correspond to simply compute the size of the product . Since has size on and on for , the product has size on . In summary, the Weil-Petersson curvature of is

On the other hand, the geodesic representative of has hyperbolic length satisfying

so that the Weil-Petersson distance of to the boundary of moduli space is

In summary, we exhibited Riemann surfaces at Weil-Petersson distance to the boundary of moduli space where some Weil-Petersson sectional curvature has size

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