Posted by: matheuscmss | June 25, 2018

Wolpert’s examples of tiny Weil-Petersson sectional curvatures

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.)

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 ${\langle\phi,\psi\rangle_{WP}}$ between two quadratic differentials ${\phi}$ and ${\psi}$ on a Riemann surface ${S}$ is

$\displaystyle \langle\phi,\psi\rangle_{WP} = \int_S \phi\overline{\psi}(ds^2)^{-1} \ \ \ \ \ (1)$

where ${ds^2}$ is the hyperbolic metric of ${S}$.

Remark 1 The quadratic differentials ${\phi}$ and ${\psi}$ are locally given by ${\phi=f(z)dz^2}$ and ${\psi=g(z)dz^2}$, so that ${\phi\psi = f(z) \overline{g(z)} dz^2d\overline{z}^2 = f(z) \overline{g(z)} |dz|^4}$. In particular, we use the hyperbolic metric to obtain a ${L^2}$-type formula (because the area form is ${|dz|^2}$).

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 ${\alpha}$ on a Riemann surface ${S}$ might have arbitrarily small hyperbolic length ${\ell_S(\alpha)}$. Moreover, the Weil-Petersson metric is incomplete because we can pinch off a hyperbolic closed geodesic ${\alpha}$ on ${S}$ in finite time ${\leq \ell_S(\alpha)^{1/2}}$. 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 ${X}$ of moduli space of Riemann surfaces. More concretely, we want to understand WP sectional curvatures ${K}$ of cotangent planes to ${X}$ in terms of the distance ${d}$ of ${X}$ to the boundary (of Deligne-Mumford compactification).

By Wolpert’s work, we know that ${-K=O(1/d)}$, i.e., WP sectional curvatures ${K}$ are bounded away from ${-\infty}$ by a polynomial function of the inverse ${1/d}$ 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 ${1/d}$. (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 ${K}$ at ${X}$ in terms of a polynomial function of the distance ${d}$ of ${X}$ to the boundary: in fact, we will see below that Wolpert constructed examples of Riemann surfaces ${X}$ where some WP sectional curvature ${K}$ behaves like ${-K\sim \exp(-1/d^2)}$.

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 ${X}$ is close to acquire a node at a curve ${\alpha}$, then we can describe an annular region ${A}$ surrounding ${\alpha}$ using a complex parameter ${t\in\mathbb{C}}$ with ${|t|\ll 1}$ and two complex coordinates ${z}$ and ${w}$ with the following properties.

The curve ${\alpha}$ separates the annular region ${A}$ into two components ${A_1}$ and ${A_2}$. The coordinate ${z}$ takes ${A}$ to ${\{|t|\leq |z|\leq 1\}}$ in such a way ${\alpha}$ is mapped to ${\{|z|=\sqrt{|t|}\}}$ and ${A_1}$ is mapped into ${\{\sqrt{|t|}\leq |z|\leq1\}}$. Similarly, the coordinate ${w}$ takes ${A}$ to ${\{|t|\leq |w|\leq 1\}}$ in such a way ${\alpha}$ is mapped to ${\{|w|=\sqrt{|t|}\}}$ and ${A_2}$ is mapped into ${\{\sqrt{|t|}\leq |w|\leq1\}}$. Furthermore, we recover the annular region ${A}$ by identifying points via the relation

$\displaystyle zw=t$

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

Remark 2 In the plumbing construction, the size ${|t|}$ of the parameter ${t}$ gives a bound on the distance of ${X}$ to the boundary of moduli space: indeed, the hyperbolic length of the geodesic representative of ${\alpha}$ is ${\sim 1/\log(1/|t|)}$. Of course, this is coherent with the idea that ${zw=0}$ describes a node.Also, the phase of ${t}$ is related to the so-called twist parameters.

Consider the plumbing construction in the figure above. It illustrates a curve ${\alpha_t}$ separating a Riemann surface ${X_t}$ of genus ${2}$ into two torii ${T_1}$ and ${T_2}$ with natural coordinates ${z}$ and ${w}$. In these coordinates, the curve ${\alpha_t}$ is ${\{z:|z|=|t|^{1/2}\} = \{w:|w|=|t|^{1/2}\}}$.

We start with the quadratic differential ${\psi_1=dz^2}$ on ${T_1}$. If we want to extend ${\psi_2}$ to ${T_2}$ and, a fortiori, ${X_t}$, then we need to understand the behavior of ${\psi_1}$ in the portion ${\{|t|^{1/2}\leq |w|\leq 1\}}$ of ${T_2}$ intersecting the annular region surrounding ${\alpha_t}$. In other words, we have to describe ${\psi_1}$ in ${w}$-coordinates.

For this sake, we recall that the definition of plumbing construction says that ${zw=t}$. Thus, ${z=t/w}$ and the formula ${dz^2 = t^2 w^{-4} dw^2}$ allows us to extend ${\psi_1}$ to ${X_t}$.

This description has the following interesting consequence: while the Weil-Petersson size of ${\psi_1}$ on ${T_1}$ is ${\sim 1}$ (because ${\psi_1|_{T_1}=dz^2}$), the Weil-Petersson size (cf. (1)) of ${\psi_1}$ on ${T_2}$ is ${\sim |t|}$ because

$\displaystyle \int_{|t|^{1/2}\leq |w|\leq 1} \underbrace{\frac{|t|^4}{|w|^8}\frac{(dw d\overline{w})^2}{|dw|^2}}_{L^2\textrm{-norm of } \psi_1=t^2w^{-4} dw^2} \underbrace{|w|^2\log^2|w|}_{(\textrm{hyperbolic metric})^{-1}}\sim \int_{|t|^{1/2}\leq |w|\leq 1} \frac{|t|^4}{|w|^6} |dw|^2$

$\displaystyle \sim \int_{\sqrt{|t|}}^1 \frac{|t|^4}{r^6}r dr\sim |t|$

By exchanging the roles of the subindices ${1}$ and ${2}$ in the previous discussion, we also get a quadratic differential ${\psi_2}$ with Weil-Petersson size ${\sim 1}$ on ${T_2}$ and ${\sim |t|}$ on ${T_1}$.

Remark 3 The 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 ${P(\psi_1, \psi_2)}$ spanned by ${\psi_1}$ and ${\psi_2}$ is tiny in the following sense.

It is explained in this paper of Wolpert that the Weil-Petersson curvature is a sort of “${L^4}$-norm” which in the case of ${P(\psi_1,\psi_2)}$ correspond to simply compute the size of the product ${\psi_1\psi_2}$. Since ${\psi_n}$ has size ${\sim 1}$ on ${T_n}$ and ${\sim |t|}$ on ${T_{3-n}}$ for ${n\in\{1,2\}}$, the product ${\psi_1\psi_2}$ has size ${\sim |t|}$ on ${X_t=T_1\cup T_2}$. In summary, the Weil-Petersson curvature of ${P(\psi_1,\psi_2)}$ is

$\displaystyle \sim-|t|$

On the other hand, the geodesic representative of ${\alpha_t}$ has hyperbolic length ${\ell(t)}$ satisfying

$\displaystyle \ell(t)\sim1/\log(1/|t|),$

so that the Weil-Petersson distance of ${X_t}$ to the boundary of moduli space is

$\displaystyle \sim\ell(t)^{1/2}\sim 1/\log^{1/2}(1/|t|)$

In summary, we exhibited Riemann surfaces ${X}$ at Weil-Petersson distance ${d\rightarrow 0}$ to the boundary of moduli space where some Weil-Petersson sectional curvature has size

$\displaystyle \sim -\exp(-1/d^2)$

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