## Polyhedral surfaces and determinant of Laplacian

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- by Alexey Kokotov PDF
- Proc. Amer. Math. Soc.
**141**(2013), 725-735 Request permission

## 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

## Additional Information

**Alexey Kokotov**- Affiliation: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montreal, Quebec, H3G 1M8 Canada
- MR Author ID: 252297
- Email: alexey@mathstat.concordia.ca
- Received by editor(s): July 5, 2010
- Published electronically: June 22, 2012
- Communicated by: Varghese Mathai
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**141**(2013), 725-735 - MSC (2010): Primary 58J52; Secondary 14H15, 30F10, 32G15
- DOI: https://doi.org/10.1090/S0002-9939-2012-11531-X
- MathSciNet review: 2996977