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For $g\geq 2$, let $M_g$ be the moduli space of genus $g$ hyperbolic surfaces, and let $\lambda_1(S_x): M_g \to \mathbb{R}$ be the smallest eigenvalue of the Laplace-Beltrami operator on the surface $S_x$ parametrized by a point $x\in M_g$. Is there anything known about how the values of $\lambda_1$ are distributed when viewed as a function on moduli space? For example, does the volume of the set of surfaces with $\lambda_1(S_x)<\varepsilon$ go to zero rapidly as $\varepsilon \to 0$?

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Volume with respect to which metric? Two metrics that people consider are the Weil-Petersson and Teichmuller metrics. –  Ian Agol Jul 1 '10 at 18:53
I have not seen too many results on Teichmuller volumes... –  Igor Rivin Dec 31 '10 at 5:29

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Not anywhere near as much as known as one might like, but for enlightenment on your specific question see M. Mirzakhani's recent preprint on arXiv.org:


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