The Journal of Geometric Analysis

, Volume 4, Issue 2, pp 207-218

First online:

Congruence subgroups and maximal Riemann surfaces

  • Paul SchmutzAffiliated withMathematisches Institut, ETH-Zentrum Email author 

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A global maximal Riemann surface is a surface of constant curvature −1 with the property that the length of its shortest simple closed geodesic is maximal with respect to all surfaces of the corresponding Teichmüller space. I show that the Riemann surfaces that correspond to the principal congruence subgroups of the modular group are global maximal surfaces. This result provides a strong geometrical reason that the Selberg conjecture, which says that these surfaces have no eigenvalues of the Laplacian in the open interval (0, 1/4), is true.

Math Subject Classification

30F 20H05 53C22

Key Words and Phrases

Riemann surfaces congruence subgroups geodesics systoles Selberg’s 1/4 conjecture