Bennington College

# Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula

 dc.contributor.author McIntyre, Andrew dc.contributor.author Takhtajan, Leon A. dc.date.accessioned 2016-12-09T18:59:21Z dc.date.available 2016-12-09T18:59:21Z dc.date.issued 2006-09 dc.identifier.citation McIntyre, A. & Takhtajan, L.A. GAFA, Geom. funct. anal. (2006) 16: 1291. doi:10.1007/s00039-006-0582-7 en_US dc.identifier.uri http://hdl.handle.net/11209/10687 dc.description.abstract For a family of compact Riemann surfaces X_t of genus g>1 parametrized by the Schottky space S_g, we define a natural basis for the holomorphic n-differentials on X_t which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n=1. We introduce a holomorphic function F(n) on S_g which generalizes the classical product \prod(1-q^m)^2 appearing in the Dedekind eta function for n=1 and g=1. We prove a holomorphic factorization formula expressing the regularized determinant of the Laplacian as a product of |F(n)|^2, a holomorphic anomaly depending on the classical Liouville action (a Kahler potential of S_g), and the determinant of the Gram matrix of the natural basis. The factorization formula reduces to Kronecker's first limit formula when n=1 and g=1, and to Zograf's factorization formula for n=1 and g>1. en_US dc.language.iso en en_US dc.publisher Springer Verlag en_US dc.subject Schottky group en_US dc.subject determinant of Laplacian en_US dc.subject Kronecker limit formula en_US dc.subject Dedekind eta function en_US dc.subject Liouville action en_US dc.subject Schottky space en_US dc.subject Teichmüller space en_US dc.subject Green’s function en_US dc.title Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula en_US dc.type Article en_US
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