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SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

Part of: Deformations of analytic structures Riemann surfaces

Published online by Cambridge University Press:  04 February 2020

FRANCISCO ARANA-HERRERA  and
JAYADEV S. ATHREYA
FRANCISCO ARANA-HERRERA
Affiliation:
Department of Mathematics, Stanford University, 450 Jane Stanford Way, Building 380, Stanford, CA 94305-2125, USA; [email protected]
JAYADEV S. ATHREYA
Affiliation:
Department of Mathematics, University of Washington, Padelford Hall, Seattle, WA 98195-4350, USA; [email protected]

Abstract

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Given integers $g,n\geqslant 0$ satisfying $2-2g-n<0$, let ${\mathcal{M}}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$ which assigns to $X\in {\mathcal{M}}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length ${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of ${\mathcal{M}}_{g,n}$ and deduce that $B$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $B$ to statistics of counting problems for simple closed hyperbolic geodesics.

MSC classification

Primary: 30F60: Teichmüller theory
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory
Type
Differential Geometry and Geometric Analysis
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e9
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Arana-Herrera, F., ‘Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani’s asymptotics for simple closed hyperbolic geodesics’, arXiv e-prints (Feb 2019), arXiv:1902.05626.Google Scholar
Arana-Herrera, F., ‘Equidistribution of horospheres on moduli spaces of hyperbolic surfaces’, (Dec 2019), arXiv:1912.03856.Google Scholar
Bers, L., ‘An inequality for Riemann surfaces’, inDifferential Geometry and Complex Analysis (Springer, Berlin, 1985), 87–93.CrossRefGoogle Scholar
Bonahon, F., ‘The geometry of Teichmüller space via geodesic currents’, Invent. Math. 92(1) (1988), 139–162.CrossRefGoogle Scholar
Bonahon, F., ‘Geodesic laminations on surfaces’, inLaminations and Foliations in Dynamics, Geometry and Topology (Stony Brook, NY, 1998), Contemporary Mathematics, 269 (American Mathematical Society, Providence, RI, 2001), 1–37.Google Scholar
Buser, P., Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, 106 (Birkhäuser Boston, Inc., Boston, MA, 1992).Google Scholar
Eskin, A., Mirzakhani, M. and Mohammadi, A., ‘Effective counting of simple closed geodesics on hyperbolic surfaces’, arXiv e-prints (May 2019), arXiv:1905.04435.Google Scholar
Farb, B. and Margalit, D., A Primer on Mapping Class Groups, Princeton Mathematical Series, 49 (Princeton University Press, Princeton, NJ, 2012).Google Scholar
Hubbard, J. H., Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 2: Surface Homeomorphisms and Rational Functions, (Matrix Editions, Ithaca, NY, 2016).Google Scholar
Martelli, B., ‘An introduction to geometric topology’. ArXiv e-prints (Oct. 2016).Google Scholar
Mirzakhani, M., ‘Random hyperbolic surfaces and measured laminations’, inIn the Tradition of Ahlfors–Bers. IV, Contemporary Mathematics, 432 (American Mathematical Society, Providence, RI, 2007), 179–198.CrossRefGoogle Scholar
Mirzakhani, M., ‘Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces’, Invent. Math. 167(1) (2007), 179–222.CrossRefGoogle Scholar
Mirzakhani, M., ‘Weil–Petersson volumes and intersection theory on the moduli space of curves’, J. Amer. Math. Soc. 20(1) (2007), 1–23.CrossRefGoogle Scholar
Mirzakhani, M., ‘Ergodic theory of the earthquake flow’, Int. Math. Res. Not. IMRN 2008(3) (2008), rnm116, https://doi.org/10.1093/imrn/rnm116.Google Scholar
Mirzakhani, M., ‘Growth of the number of simple closed geodesics on hyperbolic surfaces’, Ann. of Math. (2) 168(1) (2008), 97–125.CrossRefGoogle Scholar
Mirzakhani, M., ‘Growth of Weil–Petersson volumes and random hyperbolic surfaces of large genus’, J. Differential Geom. 94(2) (2013), 267–300.CrossRefGoogle Scholar
Penner, R. C. and Harer, J. L., Combinatorics of Train Tracks, Annals of Mathematics Studies, 125 (Princeton University Press, Princeton, NJ, 1992).CrossRefGoogle Scholar
Wolpert, S., ‘On the Weil-Petersson geometry of the moduli space of curves’, Amer. J. Math. 107(4) (1985), 969–997.CrossRefGoogle Scholar
Wolpert, S. A., Families of Riemann Surfaces and Weil–Petersson Geometry, CBMS Regional Conference Series in Mathematics, 113 (American Mathematical Society, Providence, RI, 2010), Published for the Conference Board of the Mathematical Sciences, Washington, DC.CrossRefGoogle Scholar