The length of closed geodesics on random Riemann Surfaces

2005/04/08 by Eran Makover, Makover, Eran, Jeffrey McGowan +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #math.MG #msc:58J50

paper · pdf · doi:10.48550/arxiv.math/0504175

6 figures

arxiv created 2005/04/08 · arxiv updated 2009/12/01

Abstract

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first 3g-3 geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short (< log (g)) geodesics on a random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length γ₁≤ γ₂≤ ... ≤ γ_i ,..., then for fixed k, if one allows the genus to go to infinity, the length of γ_k is independent of the genus.

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