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Approximating Orbifold Spectra Using Collapsing Connected Sums

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Abstract

For a closed Riemannian orbifold O, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain U in O whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of O can be perturbed to ensure that the first N eigenvalues of U and O are arbitrarily close to one another. This involves a generalization of the Hodge decomposition to the case of orbifolds with manifold boundary. Using these results, we study the behavior of the Laplace spectrum on functions or forms of a connected sum of two Riemannian orbifolds as one orbifold in the pair is collapsed to a point. We show that the limits of the eigenvalues of the connected sum are equal to those of the noncollapsed orbifold in the pair. In doing so, we prove the existence of a sequence of orbifolds with singular points whose eigenvalue spectra come arbitrarily close to the spectrum of a manifold, and a sequence of manifolds whose eigenvalue spectra come arbitrarily close to the eigenvalue spectrum of an orbifold with singular points. We also consider the question of prescribing the first part of the spectrum of an orientable orbifold.

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References

  1. Adem, A., Leida, J., Ruan, Y.: Orbifolds and stringy topology. Cambridge Tracts in Mathematics, 171. Cambridge University Press, Cambridge (2007)

  2. Anné, C.: Perturbation du spectre \(X \setminus TUB^\epsilon Y\) (conditions de Neumann). Séminaire de Théorie Spectrale et Géométrie, No. 4, Année 1985–1986, Univ. Grenoble I, Saint-Martin-d’Hères, 17–23 (1986)

  3. Anné, C.: Spectre du laplacien et écrasement d’anses. Ann. Sci. École Norm. Sup. (4) 20, 271–280 (1987)

    Article  MathSciNet  Google Scholar 

  4. Anné, C., Colbois, B.: Opérateur de Hodge-Laplace sur des variétés compactes privées d’un nombre fini de boules. J. Funct. Anal. 115, 190–211 (1993)

    Article  MathSciNet  Google Scholar 

  5. Anné, C., Colbois, B.: Spectre du Laplacien agissant sur let \(p\)-formes différentielles et écrasement d’anses. Math. Ann. 303, 545–573 (1995)

    Article  MathSciNet  Google Scholar 

  6. Anné, C., Post, O.: Wildly perturbed manifolds: norm resolvent and spectral convergence, preprint, arXiv:1802.01124 [math.SP], (2018)

  7. Anné, C., Takahashi, J.: \(p\)-spectrum and collapsing of connected sums. Trans. Am. Math. Soc. 364, 1711–1735 (2012)

    Article  MathSciNet  Google Scholar 

  8. Arias-Marco, T., Dryden, E.B., Gordon, C.S., Hassannezhad, A., Ray, A., Stanhope, E.: Spectral geometry of the Steklov problem on orbifolds. Int. Math. Res. Not. IMRN, 90–139 (2019)

  9. Baily Jr., W.: The decomposition theorem for V-manifolds. Am. J. Math. 78, 862–888 (1956)

    Article  MathSciNet  Google Scholar 

  10. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer, Berlin (2004)

  11. Bucicovschi, B.: Seeley’s theory of pseudodifferential operators on orbifolds, preprint, arXiv:math/9912228 [math.DG] (2008)

  12. Chavel, I., Feldman, E.A.: Spectra of domains in compact manifolds. J. Func. Anal. 30, 198–222 (1978)

    Article  MathSciNet  Google Scholar 

  13. Chavel, I., Feldman, E.A.: Spectra of manifolds less a small domain. Duke Math. J. 56, 399–414 (1988)

    Article  MathSciNet  Google Scholar 

  14. Chen, W., Ruan, Y.: A new cohomology theory of orbifold. Commun. Math. Phys. 248, 1–31 (2004)

    Article  MathSciNet  Google Scholar 

  15. Chiang, Y.-J.: Harmonic maps of \(V\)-manifolds. Ann. Glob. Anal. Geom. 8, 315–344 (1990)

    Article  MathSciNet  Google Scholar 

  16. Colbois, B., El Soufi, A.: Spectrum of the Laplacian with weights. Ann. Glob. Anal. Geom. 55, 149–180 (2019)

    Article  MathSciNet  Google Scholar 

  17. Colin de Verdière, Y.: Sur la multiplicité de la première valeur propre non nulle du laplacien. Comment. Math. Helv. 61, 254–270 (1986)

    Article  MathSciNet  Google Scholar 

  18. Colin de Verdière, Y.: Construction de laplaciens dont une partie finie du spectre est donnée. Ann. Sci. École Norm. Sup. (4) 20, 599–615 (1987)

    Article  MathSciNet  Google Scholar 

  19. Colin de Verdière, Y.: Construction de laplaciens dont une partie finie (avec multiplicités) du spectre est donnée. Séminaire sur les équations aux dérivées partielles 1986–1987, Exp. No. VII, 6 pp., École Polytech., Palaiseau (1987)

  20. Colin de Verdière, Y.: Sur une hypothèse de transversalité d’Arnold. Comment. Math. Helv. 63, 184–193 (1988)

    Article  MathSciNet  Google Scholar 

  21. Davies, E.B.: Spectral theory and differential operators. Cambridge Studies in Advanced Mathematics, 42. Cambridge University Press, Cambridge (1995)

  22. Doyle, P., Rossetti, J.P.: Isospectral hyperbolic surfaces have matching geodesics. New York J. Math. 14, 193–204 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Dryden, E., Gordon, C., Greenwald, S., Webb, D.: Asymptotic expansion of the heat kernel for orbifolds. Michigan Math. J. 56, 205–238 (2008)

    Article  MathSciNet  Google Scholar 

  24. Dryden, E., Strohmaier, A.: Huber’s theorem for hyperbolic orbisurfaces. Can. Math. Bull. 52, 66–71 (2009)

    Article  MathSciNet  Google Scholar 

  25. Dodziuk, J.: Eigenvalues of the Laplacian on forms. Proc. Am. Math. Soc. 85(3), 437–443 (1982)

    Article  MathSciNet  Google Scholar 

  26. Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI (1998)

  27. Farsi, C.: Orbifold spectral theory. Rocky Mount. J. Math. 31, 215–235 (2001)

    MathSciNet  MATH  Google Scholar 

  28. Gordon, C., Rossetti, J.P.: Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn’t reveal. Ann. Inst. Fourier (Grenoble) 53, 2297–2314 (2003)

    Article  MathSciNet  Google Scholar 

  29. Gornet, R., McGowan, J.: Small eigenvalues of the Hodge Laplacian for three-manifolds with pinched negative curvature. In: Spectral problems in geometry and arithmetic (Iowa City, IA, 1997), 29–38, Contemp. Math., 237, Amer. Math. Soc., Providence, RI (1999)

  30. Hepworth, R.: Morse inequalities for orbifold cohomology. Algebr. Geom. Topol. 9, 1105–1175 (2009)

    Article  MathSciNet  Google Scholar 

  31. Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Grundlehren der mathematischen Wissenschaften 274, Springer, Berlin (1985)

  32. Jammes, P.: Prescription de la multiplicité des valeurs propres du laplacien de Hodge-de Rham. Comment. Math. Helv. 86, 967–984 (2011)

    Article  MathSciNet  Google Scholar 

  33. Linowitz, B., Meyer, J.S.: On the isospectral orbifold-manifold problem for nonpositively curved locally symmetric spaces. Geom. Dedicata 188, 165–169 (2017)

    Article  MathSciNet  Google Scholar 

  34. McGowan, J.: The \(p\)-spectrum of the Laplacian on compact hyperbolic three manifolds. Math. Ann. 297, 725–745 (1993)

    Article  MathSciNet  Google Scholar 

  35. Post, O.: Boundary pairs associated with quadratic forms. Math. Nachr. 289, 1052–1099 (2016)

    Article  MathSciNet  Google Scholar 

  36. Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27–59 (1975)

    Article  MathSciNet  Google Scholar 

  37. Rossetti, J.P., Schueth, D., Weilandt, M.: Isospectral orbifolds with different maximal isotropy orders. Ann. Glob. Anal. Geom. 34, 351–366 (2008)

    Article  MathSciNet  Google Scholar 

  38. Sarkar, S., Suh, D.Y.: A new construction of lens spaces. Topol. Appl. 240, 1–20 (2018)

    Article  MathSciNet  Google Scholar 

  39. Satake, I.: On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. U.S.A. 42, 359–363 (1956)

    Article  MathSciNet  Google Scholar 

  40. Schwarz, G.: Hodge decomposition—a method for solving boundary value problems. Lecture Notes in Mathematics, 1607. Springer, Berlin (1995)

  41. Sutton, C.: Equivariant isospectrality and Sunada’s method. Arch. Math. (Basel) 95, 75–85 (2010)

    Article  MathSciNet  Google Scholar 

  42. Takahashi, J.: Collapsing of connected sums and the eigenvalues of the Laplacian. J. Geom. Phys. 40, 201–208 (2002)

    Article  MathSciNet  Google Scholar 

  43. Takahashi, J.: On the Gap between the First Eigenvalues of the Laplacian on Functions and \(p\)-Forms. Ann. Glob. Anal. Geom. 23, 13–27 (2003)

    Article  MathSciNet  Google Scholar 

  44. Taylor, M.: Partial Differential Equations. I. Basic Theory. Second edition. Applied Mathematical Sciences, 115. Springer, New York (2011)

  45. Wei, G.: Manifolds with a lower Ricci curvature bound. Surveys in differential geometry. Vol. XI, 203–227, Int. Press, Somerville, MA (2007)

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Acknowledgements

The authors thank Colette Anné and Junya Takahashi for helpful correspondence in the course of this work. The authors would also like to thank the anonymous referees for their very useful comments and suggestions that significantly improved this paper. C.S. and E.P. would like to thank the Department of Mathematics at the University of Colorado at Boulder, and C.F. and E.P. would like to thank the Department of Mathematics and Computer Science at Rhodes College, for hospitality during work on this manuscript. C.F. would like to thank the sabbatical program at the University of Colorado at Boulder and was partially supported by the Simons Foundation Collaboration Grant for Mathematicians #523991. E.P. would like to thank the sabbatical program at Middlebury College. C.S. would like to thank the sabbatical program at Rhodes College and was partially supported by the E.C. Ellett Professorship in Mathematics.

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado at Boulder, UCB 395, Boulder, CO, 80309-0395, USA

    Carla Farsi

  2. Department of Mathematics, Middlebury College, Middlebury, VT, 05753, USA

    Emily Proctor

  3. Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN, 38112, USA

    Christopher Seaton

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  1. Carla Farsi
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  2. Emily Proctor
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  3. Christopher Seaton
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Correspondence to Christopher Seaton.

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Farsi, C., Proctor, E. & Seaton, C. Approximating Orbifold Spectra Using Collapsing Connected Sums. J Geom Anal 31, 9433–9468 (2021). https://doi.org/10.1007/s12220-021-00611-6

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