×

Asymptotic expansion of the heat kernel for orbifolds. (English) Zbl 1175.58010

Mich. Math. J. 56, No. 1, 205-238 (2008); erratum ibid. 66, No. 1, 221-222 (2017).
Orbifolds occur naturally in many different areas of mathematics. Though they are not smooth, many results on smooth manifolds can be generalized to them. This paper studies the spectral theory of Riemannian orbifolds. Specifically, it gives a new construction of the heat kernel of a Riemannian orbifold via parametrix and obtains several leading terms in the small time asymptotic expansion of the heat kernel. By evaluating these terms for two dimensional compact orbifolds, it shows that the topological type of a two dimensional orbifold is completely determined by its spectrum (Theorem 5.15). It also shows that Riemannian orbifolds with certain restrictions on the singularities are not isospectral to a Riemannian manifold (Theorem 5.1).
The introduction of this paper gives a very good summary of results on spectral theory of compact orbifolds.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J53 Isospectrality
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. Adem and Y. Ruan, Twisted orbifold K-theory, Comm. Math. Phys. 237 (2003), 533–556. · Zbl 1051.57022
[2] M. Berger, P. Gauduchon, and E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Math., 194, Springer-Verlag, Berlin, 1971. · Zbl 0223.53034
[3] B. C. Berndt and B. P. Yeap, Explicit evaluations and reciprocity theorems for finite trigonometric sums, Adv. in Appl. Math. 29 (2002), 358–385. · Zbl 1011.11057
[4] T. P. Branson and P. B. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), 245–272. · Zbl 0721.58052
[5] J. Brüning and M. Lesch, On the spectral geometry of algebraic curves, J. Reine Angew. Math. 474 (1996), 25–66. · Zbl 0846.14018
[6] J. Brüning and R. Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), 369–429. · Zbl 0625.47040
[7] H.-W. Chen, On some trigonometric power sums, Int. J. Math. Math. Sci. 30 (2002), 185–191. · Zbl 1001.33001
[8] Y.-J. Chiang, Spectral geometry of \( V\) -manifolds and its application to harmonic maps, Differential geometry: Partial differential equations on manifolds (Los Angeles, 1990), Proc. Sympos. Pure Math., 54, pp. 93–99, Amer. Math. Soc., Providence, RI, 1993. · Zbl 0806.58005
[9] J. H. Conway, The orbifold notation for surface groups, Groups, combinatorics and geometry (Durham, 1990), London Math. Soc. Lecture Note Ser., 165, pp. 438–447, Cambridge Univ. Press, Cambridge, 1992. · Zbl 0835.20048
[10] H. Donnelly, Spectrum and the fixed point sets of isometries I, Math. Ann. 224 (1976), 161–170. · Zbl 0319.53031
[11] ——, Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23 (1979), 485–496. · Zbl 0411.53033
[12] P. Doyle and J. P. Rossetti, Isospectral hyperbolic surfaces have matching geodesics, preprint, · Zbl 1146.58026
[13] E. B. Dryden and A. Strohmaier, Huber’s theorem for hyperbolic orbisurfaces, Canad. Math. Bull. (to appear). · Zbl 1179.58014
[14] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Springer-Verlag, Berlin, 2000. · Zbl 0955.22001
[15] C. Farsi, Orbifold spectral theory, Rocky Mountain J. Math. 31 (2001), 215–235. · Zbl 0977.58025
[16] M. E. Fisher, Problem 69-14, SIAM Rev. 13 (1971), 116–119.
[17] J. B. Gil, Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr. 250 (2003), 25–57. · Zbl 1018.58014
[18] C. S. Gordon and J. P. Rossetti, Boundary volume and length spectra of Riemannian manifolds: What the middle degree Hodge spectrum doesn’t reveal, Ann. Inst. Fourier (Grenoble) 53 (2003), 2297–2314. · Zbl 1049.58033
[19] I. Moerdijk, Orbifolds as groupoids, an introduction, Orbifolds in mathematics and physics (Madison, 2001), Contemp. Math., 310, pp. 205–222, Amer. Math. Soc., Providence, RI, 2002. · Zbl 1041.58009
[20] I. Moerdijk and D. Pronk, Orbifolds, sheaves and groupoids, \(K\)-Theory 12 (1997), 3–21. · Zbl 0883.22005
[21] J. M. Montesinos, Classical tessellations and three-manifolds, Springer-Verlag, Berlin, 1987. · Zbl 0626.57002
[22] E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118 (1996), 1249–1275. · Zbl 0865.58047
[23] L. L. Pennisi, Elements of complex variables, Holt, Rinehart & Winston, New York, 1966.
[24] K. Richardson, Traces of heat operators on Riemannian foliations, · Zbl 1203.53025
[25] J. P. Rossetti, D. Schueth, and M. Weilandt, Isospectral orbifolds with different maximal isotropy orders, Ann. Global Anal. Geom. (to appear). · Zbl 1166.58016
[26] I. Satake, The Gauss–Bonnet theorem for \(V\) -manifolds, J. Math. Soc. Japan 9 (1957), 464–492. · Zbl 0080.37403
[27] N. Shams, E. A. Stanhope, and D. L. Webb, One cannot hear orbifold isotropy type, Arch. Math. (Basel) 87 (2006), 375–384. · Zbl 1111.58029
[28] E. A. Stanhope, Spectral bounds on orbifold isotropy, Ann. Global Anal. Geom. 27 (2005), 355–375. · Zbl 1085.58026
[29] E. A. Stanhope and A. Uribe, The trace formula for orbifolds, · Zbl 1223.58024
[30] W. P. Thurston, Geometry and topology of \(3\) -manifolds, Lecture notes, 1980, \(\langle \)http://www.msri.org/publications/books/gt3m/\(\rangle .\)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.