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Transactions of the American Mathematical Society

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Moduli spaces of meromorphic functions and determinant of the Laplacian


Authors: Luc Hillairet, Victor Kalvin and Alexey Kokotov
Journal: Trans. Amer. Math. Soc. 370 (2018), 4559-4599
MSC (2010): Primary 58J52, 47A10, 30F30; Secondary 14H15, 14H81, 14K25
DOI: https://doi.org/10.1090/tran/7430
Published electronically: March 19, 2018
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Abstract: 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.


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Additional Information

Luc Hillairet
Affiliation: MAPMO (UMR 7349 Université d’Orléans-CNRS) UFR Sciences, B$â$timent de mathématiques rue de Chartres, BP 6759 45067 Orléans Cedex 02, France
Email: luc.hillairet@univ-orleans.fr

Victor Kalvin
Affiliation: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulavard West, Montreal, Quebec, H3G 1M8 Canada
Email: vkalvin@gmail.com

Alexey Kokotov
Affiliation: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulavard West, Montreal, Quebec, H3G 1M8 Canada
Email: alexey.kokotov@concordia.ca

DOI: https://doi.org/10.1090/tran/7430
Received by editor(s): August 11, 2016
Published electronically: March 19, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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