Square-integrability of the Mirzakhani function and statistics of simple\n closed geodesics on hyperbolic surfaces

2019/07/14 by Francisco Arana–Herrera, Jayadev S. Athreya, Arana-Herrera, Francisco +1 · 2 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1907.06287

openalex publication_date 2019/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given integers g,n \≥ 0 satisfying 2-2g-n < 0, let \M_g,n\nbe the moduli space of connected, oriented, complete, finite area hyperbolic\nsurfaces of genus g with n cusps. We study the global behavior of the\nMirzakhani function B colon \M_g,n \→ \R_{\≥ 0} which\nassigns to X \∈ \M_g,n the Thurston measure of the set of\nmeasured geodesic laminations on X of hyperbolic length \≤ 1. We improve\nbounds of Mirzakhani describing the behavior of this function near the cusp of\n\M_g,n and deduce that B is square-integrable with respect to\nthe Weil-Petersson volume form. We relate this knowledge of B to statistics\nof counting problems for simple closed hyperbolic geodesics.\n

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• Teichmüller theory via random simple closed curves
• Volume of unit balls associated to quadratic differentials

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