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

Part of: Riemann surfaces Deformations of analytic structures

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; farana@stanford.edu
JAYADEV S. ATHREYA
Affiliation:
Department of Mathematics, University of Washington, Padelford Hall, Seattle, WA 98195-4350, USA; jathreya@uw.edu

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

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