Abstract
For a quasi-Fuchsian group Γ with ordinary set Ω, and Δ n the Laplacian on n-differentials on Γ\Ω, we define a notion of a Bers dual basis \(\phi_{1},\dotsc,\phi_{2d}\) for ker Δ n . We prove that det\(\Delta_{n}/\det \langle\phi_{j},\phi_{k}\rangle\) , is, up to an anomaly computed by Takhtajan and the second author in (Commun. Math Phys 239(1-2):183–240, 2003), the modulus squared of a holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the Selberg zeta function Z(n). This generalizes the D’Hoker–Phong formula det\(\Delta_{n}=c_{g,n}Z(n)\) , and is a quasi-Fuchsian counterpart of the result for Schottky groups proved by Takhtajan and the first author in Analysis 16, 1291–1323, 2006.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlfors L.V. (1961). Some remarks on Teichmüller’s space of Riemann surfaces. Ann. Math. 74(2): 171–191
Ahlfors L.V. (1961). Curvature properties of Teichmüller’s space. J. Anal. Math. 9: 161–176
Ahlfors, L.V.: Lectures on quasiconformal mappings, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987, With the assistance of Clifford J. Earle, Jr., Reprint of the 1966 original
Bers L. (1966). A non-standard integral equation with applications to quasiconformal mappings. Acta Math. 116: 113–134
Bers, L.: Spaces of Kleinian groups, Several Complex Variables, I. In: Proc. Conf., Univ. of Maryland, College Park, Md., 1970. Springer, Berlin, pp. 9–34 (1970)
Bers, L.: Extremal quasiconformal mappings. Advances in the Theory of Riemann Surfaces. In: Proc. Conf., Stony Brook, 1969. Princeton University Press, Princeton pp. 27–52 (1971)
Bers L. (1981). Finite dimensional Teichmüller spaces and generalizations. Bull. Am. Math. Soc. 5(2): 131–172
Büser J. (1996). The multiplier-series of a Schottky group. Math. Z. 222(3): 465–477
D’Hoker E. and Phong D.H. (1986). On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys. 104(4): 537–545
Kra I. (1972). On spaces of Kleinian groups. Comment. Math. Helv. 47: 53–69
Mcintyre, A., Takhtajan, L.: Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula. Analysis 16, 1291–1323 (2006)
Takhtajan L.A. and Teo L.-P. (2003). Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography. Commun. Math. Phys. 239(1-2): 183–240
Takhtajan, L.A., Teo, L.-P.: Weil-Petersson metric on the universal Teichmüller space I : curvature properties and Chern forms. arXiv: math.CV/0312172 (2003)
Takhtajan L.A. and Zograf P.G. (1991). A local index theorem for families of \(\overline\partial\)-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces. Commun. Math. Phys. 137(2): 399–426
Wolpert S.A. (1986). Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85(1): 119–145
Zograf, P.G.: Determinants of Laplacians, Liouville action, and an analogue of the Dedekind η–function on Teichmüller space. Unpublished manuscript (1997)
Zograf, P.G., Takhtadzhyan, L.A.: A local index theorem for families of \(\overline\partial\) -operators on Riemann surfaces. Uspekhi Mat. Nauk 42(6)(258), 133–150, 248 (1987)
Zograf, P.G., Takhtadzhyan, L.A.: On the uniformization of Riemann surfaces and on the Weil–Petersson metric on the Teichmüller and Schottky spaces. Mat. Sb. (N.S.) 132(174)(3), 304–321, 444 (1987)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mcintyre, A., Teo, LP. Holomorphic Factorization of Determinants of Laplacians using Quasi-Fuchsian Uniformization. Lett Math Phys 83, 41–58 (2008). https://doi.org/10.1007/s11005-007-0204-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-007-0204-9