Metrics of Constant Positive Curvature with Conical Singularities, Hurwitz Spaces, and Determinants of Laplacians

Abstract

Abstract. Let f : X → CP 1 be a meromorphic function of degree N with simple poles and simple critical points on a compact Riemann surface X of genus g and let m be the standard round metric of curvature 1 on the Riemann sphere CP 1 . Then the pullback f * m of m under f is a metric of curvature 1 with conical singularities of conical angles 4π at the critical points of f . We study the ζ-regularized determinant of the Laplace operator on X corresponding to the metric f * m as a functional on the moduli space of the pairs (X, f ) (i.e. on the Hurwitz space H g,N (1, . . . , 1)) and derive an explicit formula for the functional.

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].…”

“…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 the case 1) . (In order to compare (3.3) with the result in [17,18], set there z 1 = 0 and z 2 = 1/µ. )…”

“…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.…”

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

“…This is the core of the problem. For singular flat and round metrics the corresponding explicit formulas were derived with the help of separation of variables [6,9,10]. In our setting this approach does not work and we use a different idea: In a neighbourhood of x = 0 that shrinks to zero as λ → −∞ we construct an approximation for Y (λ) that is sufficiently good to find b(−∞).…”