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.

Higher Selberg zeta functions

Selberg trace formula

Selberg zeta function

Zeta-regularized determinant

Teichmüller theory

Second variation

Plurisubharmonicity

Author

Ksenia Fedosova

University of Freiburg

Julie Rowlett

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Genkai Zhang

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Annals of Global Analysis and Geometry

0232-704X (ISSN) 1572-9060 (eISSN)

Vol. 57 1 23-60

Subject Categories

Algebra and Logic

Geometry

Mathematical Analysis

DOI

10.1007/s10455-019-09687-4

More information

Latest update

2/24/2020