Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Spectral Convergence for
Degenerating Sequences of
Three Dimensional Hyperbolic Manifolds

Author: Lizhen Ji
Journal: Trans. Amer. Math. Soc. 348 (1996), 2673-2688
MSC (1991): Primary 58G25; Secondary 58C40
MathSciNet review: 1360224
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For degenerating sequences of three dimensional hyperbolic manifolds of finite volume, we prove convergence of their eigenfunctions, heat kernel and spectral measure.

References [Enhancements On Off] (What's this?)

  • 1. B. Colbois, G. Courtois, Sur les petites valeurs propres des variete hyperboliques de dimension 3, preprint, Grenoble, 1989.
  • 2. B. Colbois, G. Courtois, Convergence de varietes et convergence du spectre du Laplacien, Ann. Scient. Éc. Norm. Sup. 24 (1991) 507-518. MR 92h:53053
  • 3. I. Chavel, J. Dodziuk, The spectrum of degenerating hyperbolic manifolds of three dimensions, Journal of Diff. Geom. 39 (1994) 123-137. MR 95e:58175
  • 4. J. Dodziuk, J. McGowan, The spectrum of the Hodge Laplacian for a degenerating family of hyperbolic three manifolds, Trans. Amer. Math. Soc. 347 (1995) 1981-1995. MR 96a:58196
  • 5. J. Elstrodt, F. Grunewald, J. Mennicke, Eisenstein series on three dimensional hyperbolic space and imaginary quadratic number fields, J. reine angew. Math. 360 (1985) 160-213. MR 87c:11052
  • 6. D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math Wiss 224, Springer-Verlag, New York, 1977. MR 57:13109
  • 7. M. Gromov, Hyperbolic manifolds according to Thurston and Jørgensen, Sem. Bourbaki, vol 546, pp. 1-14, published in Springer Lecture Notes in Math, vol. 842, 1981, pp. 40-53. MR 84b:53046
  • 8. D. Hejhal, A continuity method for spectral theory on Fuchsian groups, in Modular Forms, ed. by A. Rankin, Horwood, Chichester, 1984, pp. 107-140. MR 87g:11063
  • 9. D. Hejhal, Regular $b$-groups, degenerating Riemann surfaces and spectral theory, Memoirs of Amer. Math. Soc., no. 437, 1990. MR 92h:11043
  • 10. W. Thurston, The Geometry and Topology of Three Manifolds, Lecture Notes at Princeton University, 1979.
  • 11. L. Ji, Spectral degeneration of hyperbolic Riemann surfaces, Journal of Diff. Geom. 38 (1993) 263-313. MR 94j:58172
  • 12. L. Ji, The asymptotic behavior of Green's functions for degenerating hyperbolic surfaces, Math. Z. 212 (1993) 375-394. MR 94d:58152
  • 13. L. Ji, Convergence of heat kernels for degenerating hyperbolic surfaces, Proc. of Amer. Math. Soc. 123 (1995) 639-646. MR 95c:58168
  • 14. L. Ji, Geodesic rays, potential theory, and compactifications of hyperbolic spaces, preprint 1993.
  • 15. L. Ji, S. Zelditch, Hyperbolic cusp forms and spectral simplicity on compact hyperbolic surfaces, in Geometry of the Spectrum, ed. by R.Brooks, C.Gordan and P.Perry, vol. 173 in Contemporary Math., 1994. CMP 95:02
  • 16. L. Ji, M. Zworski, The remainder estimate in spectral accumulation for degenerating hyperbolic surfaces, Journal of Functional Analysis 114 (1993) 412-420. MR 94d:58151
  • 17. J. Jorgenson, R. Lundelius, Convergence theorems for relative spectral functions on hyperbolic Riemann surfaces of finite volume, preprint, July 1992.
  • 18. T. Kubota, Elementary Theory of Eisenstein Series, John Wiley & Sons, New York, 1973. MR 55:2759
  • 19. H. Maass, Über eine neue Art von Nichtanalytischen automorphen Functionen und die Bestimmung Dirichletscher reihen Durch Funktionalgleichungen. Math. Ann. 121 (1949) 141-183. MR 11:163c
  • 20. N. Mandouvalos, Spectral theory and Eisenstein series for Kleinian groups, Proc. London Math. Soc. (3) 57 (1988) 209-238. MR 89h:58201
  • 21. W. Neumann, D. Zagier, Volumes of hyperbolic three manifolds, Topology 24 (1985) 307-332. MR 87j:57008
  • 22. E. B. Vinberg, Geometry II, Encyclopaedia of Math. Sci., vol. 29, Springer-Verlag, New York, 1993. MR 94f:53002
  • 23. H. C. Wang, Topics in totally discontinuous groups, in Symmetric Spaces, ed. by Boothby and Weiss, New York, 1972, pp. 460-485. MR 54:2879
  • 24. S. Wolpert, Asymptotics of spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys. 112 (1987) 283-315. MR 89c:58136
  • 25. S. Wolpert, Spectral limits for hyperbolic surfaces I, Invent. Math. 108 (1992) 67-89. MR 93b:58160
  • 26. S. Wolpert, Spectral limits for hyperbolic surfaces II, Invent. Math. 108 (1992) 91-129. MR 93b:58160
  • 27. S. Wolpert, Disappearance of cusp forms in special families, Ann. of Math. 139 (1994) 239-291. MR 95e:11062

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58G25, 58C40

Retrieve articles in all journals with MSC (1991): 58G25, 58C40

Additional Information

Lizhen Ji
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Keywords: Spectral convergence, degenerating sequences, hyperbolic manifolds
Received by editor(s): April 11, 1995
Additional Notes: Partially supported by NSF grant DMS 9306389 and NSF postdoctoral fellowship DMS 9407427.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society