Kokotov, Alexey (2011) On the Spectral Theory of the Laplacian on Compact Polyhedral Surfaces of Arbitrary Genus. In: Computational Approach to Riemann Surfaces. Lecture Notes in Mathematics, pp. 227-253.
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Official URL: http://dx.doi.org/10.1007/978-3-642-17413-1_8
Abstract
Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self- contained survey of their basic spectral properties, we study the zeta-regularized determinant of the Laplacian as
a functional on the moduli space of these surfaces. An explicit formula for this determinant is obtained.
Divisions: | Concordia University > Faculty of Arts and Science > Mathematics and Statistics |
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Item Type: | Book Section |
Refereed: | Yes |
Authors: | Kokotov, Alexey |
Journal or Publication: | Computational Approach to Riemann Surfaces - Lecture Notes in Mathematics |
Date: | 2011 |
Digital Object Identifier (DOI): | 10.1007/978-3-642-17413-1_8 |
ID Code: | 976818 |
Deposited By: | Danielle Dennie |
Deposited On: | 28 Jan 2013 20:54 |
Last Modified: | 18 Jan 2018 17:43 |
References:
1.Aurell, E., Salomonson, P.: Further Results on Functional Determinants of Laplacians in Simplicial Complexes, hep-th/94051402.Bobenko, A.I.: Introduction to compact Riemann surfaces. In: Bobenko, A.I., Klein, Ch. (eds.) Lecture Notes in Mathematics 2013, pp. 3–64. Springer, Berlin (2011)
3.Burghelea, D., Friedlander, L., Kappeler, T.: Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107, 34–65 (1992)»
4.Carslaw, H.S.: The Green’s function for a wedge of any angle, and other problems in the conduction of heat. Proc. Lond. Math. Soc. 8, 365–374 (1910)»
5.Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Diff. Geom. 18, 575–657 (1983)
6.D’Hoker, E., Phong, D.H.: Functional determinants on Mandelstam diagrams. Comm. Math. Phys. 124, 629–645 (1989)»
7.Fay, J.D.: Theta-functions on Riemann surfaces. Lect. Notes in Math., vol. 352. Springer, Berlin (1973)
8.Fay, J.D.: Kernel functions, analytic torsion, and moduli spaces. Memoir. AMS 464 (1992)
9.Fursaev, D.V.: The heat-kernel expansion on a cone and quantum fields near cosmic strings. Class. Quant. Grav. 11, 1431–1443 (1994)»
10.Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1978)
11.King, h.K.: Determinants of Laplacians on the space of conical metrics on the sphere. Trans. AMS 339, 525–536 (1993)
12.Klochko, Y., Kokotov, A.: Genus one polyhedral surfaces, spaces of quadratic differentials on tori and determinants of Laplacians. Manuscripta Math. 122, 195–216 (2007)»
13.Kokotov, A., Korotkin, D.: Tau-functions on the spaces of Abelian and quadratic differentials and determinants of Laplacians in Strebel metrics of finite volume, preprint of the Max-Planck Institute for Mathematics in the Science, Leipzig, 46/2004; math.SP/0405042
14.Kokotov, A.: Preprint (127) of Max-Planck-Institut für Mathematik in Bonn (2007)
15.Kokotov, A., Korotkin, D.: Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray-Singer formula. J. Differ. Geom. 82, 35–100 (2009)
16.Kondratjev, V.: Boundary value problems for elliptic equations in domains with conical and angle points. Proc. Moscow Math. Soc. 16, 219–292 (1967)
17.Kontsevich, M., Zorich, A.: Connected components of the moduli spaces of holomorphic differentials with prescribed singularities. Invent. Math. 153, 631–678 (2003)» CrossRef
18.Kontsevitch, M., Zorich A.: Lyapunov exponents and Hodge theory, hep-th/9701164
19.Loya, P., McDonald, P., Park, J.: Zeta regularized determinants for conic manifolds. J. Funct. Anal. 242(1), 195–229 (2007)»
20.McKean, H.P., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom. 1, 43–69 (1967)
21.Mooers, E.: Heat kernel asymptotics on manifolds with conic singularities. J. D’Analyse Mathématique, 78, 1–36 (1999)
22.Nazarov, S., Plamenevskii, B.: Elliptic boundary value problems in domains with piece-wise smooth boundary, Moscow, “Nauka” (1992)
23.Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of laplacian. J. Funct. Anal. 80(1), 148–211 (1988)»
24.Ray D.B., Singer I.M.: Analytic torsion for complex manifolds. Ann. Math. 98(1), 154–177 (1973)»
25.Taylor, M.: Partial Differential Equations, vol. 2. Springer, New York (1996) (Appl. Math. Sci., 116)
26.Troyanov, M.: Les surfaces euclidiennes à singularités coniques. L’Enseignement Mathématique, 32, 79–94 (1986)
27.Zorich, A.: Flat Surfaces. In Cartier, P., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in Number theory, Physics and Geometry. Vol. 1: On random matrices, zeta functions, and dynamical systems, pp. 439–586. Springer, Berlin (2006)
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