Second variation of Selberg zeta functions and curvature asymptotics

Abstract

We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmüller space. We then use this formula to determine the asymptotic behavior as $\text{Re}(s)\to \infty$ of the second variation. As a consequence, for $m\in \mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t,{\mathcal{K}}_t)\to t\in \mathcal{T}$ of holomorphic m-differentials over the Teichmüller space $\mathcal{T}$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2{e}^{-l_{0}m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.

35 pages

Cited by in corpus (1)

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