Second variation of Selberg zeta functions and curvature asymptotics

Journal article, 2020

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 Rs→ ∞ of the second variation. As a consequence, for m∈ N, we obtain the complete expansion in m of the curvature of the vector bundle H(Xt, Kt) → t∈ T of holomorphic m-differentials over the Teichmüller space T, for m large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, O(m2e-l0m), where l is the length of the shortest closed hyperbolic geodesic.