Details

Autor(en) / Beteiligte

• HILLAIRET, LUC
• KALVIN, VICTOR
• KOKOTOV, ALEXEY

Titel

Moduli spaces of meromorphic functions and determinant of the Laplacian

Ist Teil von

• Transactions of the American Mathematical Society, 2018-07, Vol.370 (7), p.4559-4599

Ort / Verlag

American Mathematical Society

Erscheinungsjahr

2018

Quelle

EZB Electronic Journals Library

Beschreibungen/Notizen

• The Hurwitz space is the moduli space of pairs (X,f) where X is a compact Riemann surface and f is a meromorphic function on X. We study the Laplace operator \Delta ^{\vert df\vert^2} of the flat singular Riemannian manifold (X,\vert df\vert^2). We define a regularized determinant for \Delta ^{\vert df\vert^2} and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime-form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential df. The proof has several parts that can be of independent interest. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplace operator on flat surfaces with conical singularities and Euclidean or conical ends. We introduce and study the S-matrix, S(\lambda ), of a surface with conical singularities as a function of the spectral parameter \lambda and relate its behavior at \lambda =0 with the Schiffer projective connection on the Riemann surface X. We also prove variational formulas for eigenvalues of the Laplace operator of a compact surface with conical singularities when the latter move.