2014/10/12 by Luc Hillairet, Hillairet, Luc, Victor Kalvin +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1410.3106
openalex publication_date 2014/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Δ|df|2 of the flat singular Riemannian manifold (X,|df|2). We define a regularized determinant for Δ|df|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(λ), of a surface with conical singularities as a function of the spectral parameter λ and relate its behavior at λ=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.