Second variation of Selberg zeta functions and curvature asymptotics

TL;DR

This paper derives an explicit second variation formula for the Selberg zeta function on Teichmüller space, analyzes its asymptotic behavior, and explores the curvature of holomorphic differential bundles, revealing exponential decay in the difference from Quillen curvature.

Contribution

It provides the first explicit second variation formula for the Selberg zeta function and connects curvature asymptotics with hyperbolic geometry and Quillen curvature.

Findings

Explicit second variation formula for the Selberg zeta function.

Asymptotic expansion of curvature for large m.

Curvature difference from Quillen curvature decays exponentially.

Abstract

We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichm\"uller 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 Teichm\"uller 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.