On the asymptotics of determinant of Laplacian at the principal boundary of the principal stratum of the moduli space of Abelian differentials
Author:
A. Kokotov
Journal:
Trans. Amer. Math. Soc. 364 (2012), 5645-5671
MSC (2010):
Primary 58J52; Secondary 32G15, 14H15, 30F10
DOI:
https://doi.org/10.1090/S0002-9947-2012-05695-9
Published electronically:
June 14, 2012
MathSciNet review:
2946925
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a translation surface of genus
with
conical points of angle
and let
,
be two homologous saddle connections of length
joining two conical points of
and bounding two surfaces
and
with boundaries
and
. Gluing the opposite sides of the boundary of each surface
,
one gets two (closed) translation surfaces
,
of genera
,
;
. Let
,
and
be the Friedrichs extensions of the Laplacians corresponding to the (flat conical) metrics on
,
and
respectively. We study the asymptotical behavior of the (modified, i.e. with zero modes excluded) zeta-regularized determinant
as
and
shrink. We find the asymptotics




- 1. Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911
- 2. 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
- 3. 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, https://doi.org/10.1016/0022-1236(92)90099-5
- 4. Jeff Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), no. 4, 575–657 (1984). MR 730920
- 5. Julian Edward and Siye Wu, Determinant of the Neumann operator on smooth Jordan curves, Proc. Amer. Math. Soc. 111 (1991), no. 2, 357–363. MR 1031662, https://doi.org/10.1090/S0002-9939-1991-1031662-0
- 6. Eskin A., Masur H., Zorich A., Moduli spaces of Abelian differentials: the principal boundary, counting problems and the Siegel-Veech constants, math.DS/0202134.
- 7. John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR 0335789
- 8. John Fay, Kernel functions, analytic torsion, and moduli spaces, Mem. Amer. Math. Soc. 96 (1992), no. 464, vi+123. MR 1146600, https://doi.org/10.1090/memo/0464
- 9. Hillairet L., Contribution d'orbites périodiques diffractives à la formule de trace, Ph.D. Thesis, L'Institut Fourier, Grenoble, 2002.
- 10. Jay Jorgenson, Asymptotic behavior of Faltings’s delta function, Duke Math. J. 61 (1990), no. 1, 221–254. MR 1068387, https://doi.org/10.1215/S0012-7094-90-06111-3
- 11. 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
- 12. 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 Sciences, Leipzig, 46/2004; math.SP/0405042.
- 13. 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, https://doi.org/10.1007/s00222-003-0303-x
- 14. Yoonweon Lee, Burghelea-Friedlander-Kappeler’s gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion, Trans. Amer. Math. Soc. 355 (2003), no. 10, 4093–4110. MR 1990576, https://doi.org/10.1090/S0002-9947-03-03249-5
- 15. Paul Loya, Patrick McDonald, and Jinsung Park, Zeta regularized determinants for conic manifolds, J. Funct. Anal. 242 (2007), no. 1, 195–229. MR 2274020, https://doi.org/10.1016/j.jfa.2006.04.014
- 16. Rolf E. Lundelius, Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume, Duke Math. J. 71 (1993), no. 1, 211–242. MR 1230291, https://doi.org/10.1215/S0012-7094-93-07109-8
- 17. Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
- 18. Shin Ozawa, The first eigenvalue of the Laplacian on two-dimensional Riemannian manifolds, Tohoku Math. J. (2) 34 (1982), no. 1, 7–14. MR 651702, https://doi.org/10.2748/tmj/1178229304
- 19. Joseph Polchinski, Evaluation of the one loop string path integral, Comm. Math. Phys. 104 (1986), no. 1, 37–47. MR 834480
- 20. D. B. Ray and I. M. Singer, Analytic torsion for complex manifolds, Ann. of Math. (2) 98 (1973), 154–177. MR 383463, https://doi.org/10.2307/1970909
- 21. William I. Weisberger, Conformal invariants for determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 4, 633–638. MR 910583
- 22. Richard A. Wentworth, Precise constants in bosonization formulas on Riemann surfaces. I, Comm. Math. Phys. 282 (2008), no. 2, 339–355. MR 2421480, https://doi.org/10.1007/s00220-008-0560-z
- 23. Wentworth R., private communication.
- 24. Richard Wentworth, Asymptotics of determinants from functional integration, J. Math. Phys. 32 (1991), no. 7, 1767–1773. MR 1112704, https://doi.org/10.1063/1.529239
- 25. R. Wentworth, The asymptotics of the Arakelov-Green’s function and Faltings’ delta invariant, Comm. Math. Phys. 137 (1991), no. 3, 427–459. MR 1105425
- 26. Scott A. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 2, 283–315. MR 905169
- 27. Akira Yamada, Precise variational formulas for abelian differentials, Kodai Math. J. 3 (1980), no. 1, 114–143. MR 569537
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Additional Information
A. Kokotov
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8
Email:
alexey@mathstat.concordia.ca
DOI:
https://doi.org/10.1090/S0002-9947-2012-05695-9
Received by editor(s):
July 6, 2010
Published electronically:
June 14, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.