Title: 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 $s \to \infty$ of the second variation.
As a consequence, we determine the signature of the Hessian of $\log Z(s)$ for
sufficiently large $s$. As a further consequence, the asymptotic behavior of
the second variation of $\log Z(s)$ shows that the Ricci curvature of the Hodge
bundle $H^0(\mathcal K^m_t)\mapsto t$ over Teichmüller space agrees with the
Quillen curvature up to a term of exponential decay, $O(s^2 e^{-l_0 s}),$ where
$l_0$ is the length of the shortest closed hyperbolic geodesic.