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Weil-Petersson geometry (II)

Reformulations

Random geodesics and geodesic currents

Given two different hyperbolic metrics m and m’ on a closed topological surface, i.e. two different points in Teichmüller space, Thurston defined a quantity A(m, m') which can be interpreted as the length of a “random geodesic” in one metric measured in the other metric. A more precise definition involves intersection forms and Liouville measures and the transverse measure L_m to the geodesic flow, terms (some of) which are to be defined presently.

By the (rather computational) work of Wolpert, this metric turns out to be equivalent to the Weil-Petersson metric.

Bonahon reinterpreted this, yet again, in the language of geodesic currents. A geodesic current associated to a closed hyperbolic surface S is a measure on the set G(\tilde{S}) of unoriented geodesics of the universal cover \tilde{S} which is invariant under the natural action of \pi_1(S).

The space of geodesics G(\tilde{S}) embeds into the space of currents \mathscr{C}(S) as atomic measures, and this can be extended to an embedding of the measured laminations \mathscr{ML}(S) into \mathscr{C}(S). Indeed the image of the geodesics under scalar multiplication, i.e. \mathbb{R} \cdot G(\tilde{S}), is dense in \mathscr{C}(S).

Moreover, we can extend the geometric intersection number on closed geodesics to a continuous symmetric bilinear form i: \mathscr{C}(S) \times \mathscr{C}(S) \to \mathbb{R}^+.

What makes \mathscr{C}(S) an interesting object to consider is that there is also a natural embedding of Teichmüller space \mathcal{T}(S) \curvearrowright \mathscr{C}(S), which works by taking any point into the Liouville measure induced by the associated hyperbolic metric. This embedding is as nice as one might want it to be; in particular, its image can be characterized by analytic equations.

We can now use the intersection form to define a (path) metric on Teichmüller space, or rather on its image in the space of currents, relate this to Thurston’s metric, and then appeal to Wolpert’s result to find that this metric is in fact equivalent to the Weil-Petersson metric.

The thermodynamic formalism and the pressure metric

Yet another reinterpretation of—i.e. yet another way of defining a metric on Teichmüller space which turns out to be equivalent to—the Weil-Petersson metric involves the thermodynamic formalism from dynamics.

The thermodynamic formalism, for our purposes, is a machine from symbolic dynamics which produces analytically varying numerical invariants associated to families of (sufficiently nice) dynamical systems. In our case the dynamical system in question will be the geodesic flow on the unit tangent bundle of our closed hyperbolic surface, which may be represented as a finite-type shift using the Bowen-Series coding; we obtain our family of systems by considering Hölder reparametrizations of this flow.

From this system we obtain a space of \alpha-Hölder functions, which we view as reparametrizations of our geodesic flow, and thence a function C^{1+\alpha}(S^1) \to \mathbb{R}_{\geq 0}  the pressure function, which is equal to the log of the spectral radius of the transfer operator. Pressure-zero functions correspond to ergodic, flow-invariant equilibrium measures.

We may verify that pressure varies analytically, and so the second derivative of the pressure is well-defined, and is equal to the variance for a pressure-zero Hölder function. We may further check that the corresponding pressure metric defined by \|g\|^2_{\mathbf{P}} := \frac{\partial^2}{\partial t^2} \big|_{t=0} \mathbf{P}(f+tg) on the space of pressure-zero functions is positive-definite.

Working through this machinery, McMullen (building on the work of Bridgeman and Taylor, who extended the Weil-Petersson metric to quasifuchsian space) showed that the pressure metric may be expressed in terms of the Hessian of the Hausdorff dimension of the limit set, or of the pushforward measure on the boundary circle.

With more work, this Hausdroff dimension can also be related to the Weil-Petersson metric, and this can be used to show that the pressure metric on Teichmüller space is equivalent to the Weil-Petersson metric.

This pressure metric was subsequently extendedby Bridgeman, Canary, Labourie and Sambarino to more general higher Teichmüller spaces / spaces of Anosov representations. In that setting the technicalities are considerably more formidable—one has to produce a flow space to replace the geodesic flow on the unit tangent bundle, showing that the resulting pressure metric is well-defined and non-degenerate is much more work, etc.

Example here

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Applications

Wolpert’s expansions and estimates on curvature, together with the geometry of the boundary strata, can be used to produce an arithmetic Riemann-Roch formula for pointed stable curves, which yields e.g. the exact formula for the Selberg zeta function on the level-2 principal congruence subgroup \Gamma(2) < \mathrm{SL}_2\mathbb{R}. I don’t understand this at all well enough to comment further, but I’m going to speculate that the Weil-Petersson geometry plays the role which e.g. moduli space / stack arguments play in the Arakelov theory version.

A different (line of) application(s) appears in the groundbreaking work of Maryam Mirzakhani, who used a recursion formula for the Weil-Petersson volume of moduli space(s) to obtain asymptotic counts of simple closed geodesics on hyperbolic surfaces, establish relations to intersection theory on stable curves (including a new proof of the Witten-Kontsevich formula), applications to mathematical physics, and so on …

All of these results can also be used to study—going back to where it all began, in some sense—the properties of random Riemann surfaces of high genus.

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