Polyhedral surfaces and determinant of Laplacian

Kokotov, Alexey

doi:10.1090/S0002-9939-2012-11531-X

Polyhedral surfaces and determinant of Laplacian
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by Alexey Kokotov

Proc. Amer. Math. Soc. 141 (2013), 725-735

DOI: https://doi.org/10.1090/S0002-9939-2012-11531-X

Published electronically: June 22, 2012

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Abstract:

An explicit formula for the determinant of the Laplacian on a compact polyhedral surface of genus $g>1$ is found. This formula generalizes previously known results for flat surfaces with trivial holonomy and compact polyhedral tori.

References

Orlando Alvarez, Theory of strings with boundaries: fluctuations, topology and quantum geometry, Nuclear Phys. B 216 (1983), no. 1, 125–184. MR 701643, DOI 10.1016/0550-3213(83)90490-X
Erik Aurell and Per Salomonson, On functional determinants of Laplacians in polygons and simplicial complexes, Comm. Math. Phys. 165 (1994), no. 2, 233–259. MR 1301847, DOI 10.1007/BF02099770
D. Burghelea, L. Friedlander, and T. Kappeler, Meyer-Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992), no. 1, 34–65. MR 1165865, DOI 10.1016/0022-1236(92)90099-5
Jeff Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), no. 4, 575–657 (1984). MR 730920
John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
John Fay, Kernel functions, analytic torsion, and moduli spaces, Mem. Amer. Math. Soc. 96 (1992), no. 464, vi+123. MR 1146600, DOI 10.1090/memo/0464
Hala Khuri King, Determinants of Laplacians on the space of conical metrics on the sphere, Trans. Amer. Math. Soc. 339 (1993), no. 2, 525–536. MR 1102890, DOI 10.1090/S0002-9947-1993-1102890-7
Yulia Klochko and Alexey Kokotov, Genus one polyhedral surfaces, spaces of quadratic differentials on tori and determinants of Laplacians, Manuscripta Math. 122 (2007), no. 2, 195–216. MR 2295112, DOI 10.1007/s00229-006-0063-1
Kokotov, A., On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials, to appear in Transactions of AMS.
Aleksey Kokotov and Dmitry Korotkin, Tau-functions on spaces of abelian differentials and higher genus generalizations of Ray-Singer formula, J. Differential Geom. 82 (2009), no. 1, 35–100. MR 2504770
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 Max-Planck Institute for Mathematics in the Science, Leipzig, 46/2004; math.SP/0405042.
Maxim Kontsevich and Anton Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), no. 3, 631–678. MR 2000471, DOI 10.1007/s00222-003-0303-x
Paul Loya, Patrick McDonald, and Jinsung Park, Zeta regularized determinants for conic manifolds, J. Funct. Anal. 242 (2007), no. 1, 195–229. MR 2274020, DOI 10.1016/j.jfa.2006.04.014
H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739
B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148–211. MR 960228, DOI 10.1016/0022-1236(88)90070-5
D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177. MR 383463, DOI 10.2307/1970909
M. Spreafico, Zeta function and regularized determinant on a disc and on a cone, J. Geom. Phys. 54 (2005), no. 3, 355–371. MR 2139088, DOI 10.1016/j.geomphys.2004.10.005
Marc Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. (2) 32 (1986), no. 1-2, 79–94 (French). MR 850552
William I. Weisberger, Conformal invariants for determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 4, 633–638. MR 910583, DOI 10.1007/BF01225377
Anton Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin, 2006, pp. 437–583. MR 2261104, DOI 10.1007/978-3-540-31347-2_{1}3