[Paper Review] Equidistribution of horospheres on moduli spaces of hyperbolic surfaces

Francisco Arana–Herrera|arXiv (Cornell University)|Dec 9, 2019

Geometry and complex manifolds20 references4 citations

TL;DR

This paper generalizes Mirzakhani's equidistribution result for simple closed geodesics on hyperbolic surfaces by proving the equidistribution of horospheres in the moduli space of hyperbolic structures on surfaces of negative Euler characteristic. Using the dynamics of the earthquake flow and geometric measure-theoretic techniques, it establishes that horoball segments defined by bounded, compactly supported functions on multi-curves equidistribute with respect to the Mirzakhani measure as lengths tend to infinity.

ABSTRACT

Given a simple closed curve $γ$ on a connected, oriented, closed surface $S$ of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on $S$ having a simple closed geodesic of length $L$ of the same topological type as $γ$ equidistributes with respect to a natural probability measure as $L o \infty$. We prove several generalizations of Mirzakhani's result and discuss some of the technical aspects ommited in her original work. The dynamics of the earthquake flow play a fundamental role in the arguments in this paper.

Motivation & Objective

• To extend Mirzakhani's equidistribution result for simple closed geodesics to more general horospheric families defined by multi-curves and bounded functions.
• To fill in technical gaps in Mirzakhani's original proof concerning the behavior of horoball segments and their measures in Teichmüller space.
• To establish uniform bounds on the growth of measures associated with measured geodesic laminations in relation to the Weil-Petersson volume form.
• To leverage the earthquake flow's ergodicity and strong non-divergence to prove equidistribution results in non-locally symmetric, non-homogeneous spaces like moduli spaces of Riemann surfaces.
• To provide a rigorous foundation for future applications in counting problems for filling geodesics and related dynamical systems on moduli spaces.

Proposed method

• Define horoball segments $ B_{ u}^{f,L} o ext{Teich}(S_{g,n}) $ as level sets of hyperbolic lengths of a tuple of disjoint simple closed curves $ u = ( u_1, \\/dots, \nu_k) $, scaled by $ L $ and supported on the function $ f $.
• Construct a natural horoball segment measure $ \mu_{\nu}^{f,L} $ on $ B_{\nu}^{f,L} $, integrating the Weil-Petersson measure over the base and a coned-off version of the Thurston measure on fibers.
• Utilize the earthquake flow's ergodicity and strong non-divergence properties to analyze the asymptotic behavior of these measures as $ L \to \infty $.
• Apply geometric measure theory by introducing a flow datum $ (K(\epsilon_0), K(2\epsilon_0), F^\lambda) $ associated with the length functional $ \ell_\lambda $, using a vector field $ F^\lambda $ to generate a one-parameter diffeomorphism $ \varphi_t^\lambda $.
• Establish uniform bounds on the Jacobian determinant $ |\text{det}(\varphi_t^\lambda)| $ and fiberwise norms to control measure distortion under the flow.
• Use the identity $ \mu_{\text{wp}}(U_X(2\epsilon)) = \int_0^\infty \eta_\lambda^r(U_X(2\epsilon)) \, dr $ to relate the Weil-Petersson volume to the pushforward of the horospherical measure $ \eta_\lambda^r $, leading to a quantitative bound on $ \eta_\lambda^1(V) $.

Experimental results

Research questions

• RQ1 How can Mirzakhani's equidistribution result for simple closed geodesics be generalized to horospheres defined by multi-curves and bounded functions?
• RQ2 What role does the earthquake flow play in establishing equidistribution in non-locally symmetric, non-homogeneous moduli spaces?
• RQ3 Can uniform bounds on the growth of horospherical measures be derived using geometric flows and measure distortion estimates?
• RQ4 How do the asymptotic behaviors of horoball segments relate to the Mirzakhani measure and the Weil-Petersson volume form?
• RQ5 What technical refinements are required to complete the equidistribution proof beyond the qualitative ergodicity arguments in Mirzakhani's original work?

Key findings

• The horoball segment measures $ \mu_\nu^{f,L} $ equidistribute with respect to the Mirzakhani measure on $ P^1\mathcal{M}_{g,n} $ as $ L \to \infty $, generalizing Mirzakhani's result to multi-curves and bounded functions.
• A uniform upper bound $ \eta_\lambda^1(U_X(\epsilon)) \leq C \cdot \epsilon^{6g-6+2n-1} $ is established for all $ X \in K $ and $ \lambda \in \mathcal{ML}_{g,n}(K(\epsilon_0)) $, with $ C $ depending only on the topology.
• The function $ C(\lambda) = \frac{C_1 \cdot 2^{6g-6+2n} \cdot C_2(\lambda)}{C_3(\lambda)} $ is continuous and achieves a maximum on the compact set $ \mathcal{ML}_{g,n}(K(\epsilon_0)) $, ensuring uniform control.
• The proof relies on the flow $ \varphi_t^\lambda $ induced by a vector field $ F^\lambda $, which preserves the structure of horospheres and allows control over measure distortion via the Jacobian determinant.
• The equidistribution result is achieved by combining the integral identity $ \mu_{\text{wp}}(U_X(2\epsilon)) = \int_0^\infty \eta_\lambda^r(U_X(2\epsilon)) \, dr $ with lower bounds on $ \eta_\lambda^{1+t}(\varphi_t^\lambda(V)) $, leading to a quantitative bound on the initial measure $ \eta_\lambda^1(V) $.
• The key technical contribution is the rigorous treatment of measure distortion and volume growth in the Teichmüller space, filling a gap in Mirzakhani's original argument concerning the behavior of horoball segments near the thin part of the moduli space.