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 mathfrak {R}s rightarrow 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)rightarrow tin {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 mathrm{e}^{-l_0 m}), where l_0 is the length of the shortest closed hyperbolic geodesic.