On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

Abstract

Let m be any conical (or smooth) metric of finite volume on the Riemann sphere CP 1 . On a compact Riemann surface X of genus g consider a meromorphic funciton f : X → CP 1 such that all poles and critical points of f are simple and no critical value of f coincides with a conical singularity of m or {∞}. The pullback f * m of m under f has conical singularities of angles 4π at the critical points of f and other conical singularities that are the preimages of those of m. We study the ζ-regularized determinant Det ′ ∆ F of the (Friedrichs extension of) Laplace-Beltrami operator on (X, f * m) as a functional on the moduli space of pairs (X, f ) and obtain an explicit formula for Det ′ ∆

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

“…In the second part of the paper we demonstrate how the results of the first part can be used to obtain explicit formulas for determinants of Laplacians on singular surfaces with or without boundary, this occupies Section 3. In Subsection 3.1 we consider the constant curvature spheres with two conical singularities: we recover and generalize the corresponding results in [36,23,17,18] and discuss extremal properties of the determinant for the metrics of area 4π. In Subsection 3.2 we consider polyhedral surfaces with spherical topology and obtain an analog of the Aurell-Salomonson formula in [3].…”

“…Here, for instance, results in [2,3] can be interpreted as a generalization of Polyakov-Alvarez formula to the case of flat conical metrics on a disk and on a sphere, the main result in [21] is a simple consequence of an analog of Polyakov formula for two conformally equivalent flat conical metrics and the results in [22]. Some results were also obtained for determinants of Laplacians in constant positive curvature (spherical) [10,36,23,18,19] and other conical metrics [17], but no Polyakov-Alvarez type formulas for metrics other than smooth or conical flat were available until now.…”

“…The case µ = 0 (a sphere with two antipodal conical singularities of angle 2π(β + 1), or, equivalently, a spindle or an american football) was previously studied in [36] by an approach based on separation of variables; see also [23]. A variational formula for ζ (0) with respect to µ (for β = 1) was recently obtained in [17,18]. In the case 1) .…”

“…In Subsection 3.3 we deduce a formula for the determinant of Friederichs Dirichlet Laplacian on the constant curvature metric disks with a conical singularity at the center (for the spherical and hyperbolic metrics the result is new, the case of flat conical metrics was studied in [35]). In Subsection 3.4 we study the determinant of Friederichs Laplacian on hyperbolic spheres: we obtain a new explicit formula for the determinant, recover a variational formula from [17] and find the corresponding undetermined constant. Finally, in Subsection 3.5 we present a general explicit formula for the determinant of Friederichs Laplacian on singular genus g > 1 surface without boundary, this is a generalization of the results in [21,22].…”

Determinant of the Laplacian on Tori of Constant Positive Curvature with one Conical Point

“…Observe that the right hand side in (3.3) is actually the value of ∂ w log ρ(w,w) −1/4 at w = t, where ρ(w,w) is the conformal factor of the metric (2.2); this is also a direct consequence of [8,Lemma 4]. Using (3.1) together with (3.2) and (3.3), we are now able to derive an explicit formula for det∆ µ * m .…”