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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asked Jul 1, 2010 at 17:47
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$\begingroup$ Volume with respect to which metric? Two metrics that people consider are the Weil-Petersson and Teichmuller metrics. $\endgroup$– Ian AgolCommented Jul 1, 2010 at 18:53
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$\begingroup$ I have not seen too many results on Teichmuller volumes... $\endgroup$– Igor RivinCommented Dec 31, 2010 at 5:29
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1 Answer
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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:
answered Dec 31, 2010 at 2:42