Skip to main content
SpringerLink
Log in

Weyl laws on open manifolds

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is purely discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ammann B., Lauter R., Nistor V., Vasy A. (2004). Complex powers and non-compact manifolds. Comm. Partial Diff. Eq. 29(5–6): 671–705

    Article  MATH  MathSciNet  Google Scholar 

  2. Anghel N. (1996). Generic vanishing for harmonic spinors of twisted Dirac operators. Proc. Am. Math. Soc. 124: 3555–3561

    Article  MATH  MathSciNet  Google Scholar 

  3. Atiyah M.F., Patodi V.K., Singer I.M. (1975). Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc. 77: 43–69

    Article  MATH  MathSciNet  Google Scholar 

  4. Bär C. (2000). The Dirac operator on hyperbolic manifolds of finite volume. J. Diff. Geom. 54: 439–488

    MATH  Google Scholar 

  5. Bär C., Dahl M. (2002). Surgery and the spectrum of the Dirac operator. J. Reine Angew. Math. 552: 53–76

    MATH  MathSciNet  Google Scholar 

  6. Bär C., Gauduchon P., Moroianu A. (2005). Generalized cylinders in semi-Riemannian and spin geometry. Math. Z. 249(3): 545–580

    Article  MATH  MathSciNet  Google Scholar 

  7. Bucicovschi B. (1999). An extension of the work of V. Guillemin on complex powers and zeta functions of elliptic pseudodifferential operators. Proc. Am. Math. Soc. 127: 3081–3090

    Article  MATH  MathSciNet  Google Scholar 

  8. Christiansen T., Zworski M. (1995). Spectral asymptotics for manifolds with cylindrical ends. Ann. Inst. Fourier (Grenoble) 45: 251–263

    MATH  MathSciNet  Google Scholar 

  9. Delange H. (1954). Généralization du théorème de Ikehara. Ann. Sci. Ecole Norm. Sup. (3) 71: 213–242

    MATH  MathSciNet  Google Scholar 

  10. Friedrich T. (1998). On the spinor representation of surfaces in Euclidean 3-space. J. Geom. Phys. 28: 143–157

    Article  MATH  MathSciNet  Google Scholar 

  11. Guillemin V. (1985). A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues. Adv. Math. 55: 131–160

    Article  MATH  MathSciNet  Google Scholar 

  12. Hitchin N. (1974). Harmonic spinors. Adv. Math. 14: 1–55

    Article  MATH  MathSciNet  Google Scholar 

  13. Hörmander L. (1968). The spectral function of an elliptic operator. Acta Math. 121: 193–218

    Article  MATH  MathSciNet  Google Scholar 

  14. Ikehara S. (1931). An extension of Landau’s theorem in the analytic theory of numbers. J. Math. Phys. MIT 10: 1–12

    Google Scholar 

  15. Lauter R., Moroianu S. (2002). The index of cusp operators on manifolds with corners. Ann. Global Anal. Geom. 21: 31–49

    Article  MATH  MathSciNet  Google Scholar 

  16. Lauter R., Moroianu S. (2003). Homology of pseudodifferential operators on manifolds with fibered cusps. Trans. Am. Math. Soc. 355: 3009–3046

    Article  MATH  MathSciNet  Google Scholar 

  17. Lauter R., Moroianu S. (2005). An index formula on manifolds with fibered cusp ends. J. Geom. Anal. 15(2): 261–283

    MATH  MathSciNet  Google Scholar 

  18. Lesch M., Peyerimhoff N. (1998). On index formulas for manifolds with metric horns. Comm. Partial Diff. Eq. 23: 649–684

    MATH  MathSciNet  Google Scholar 

  19. Loya P. (2001). Tempered operators and the heat kernel and complex powers of elliptic pseudodifferential operators. Comm. Partial Diff. Eq. 26: 1253–1321

    Article  MATH  MathSciNet  Google Scholar 

  20. Maier S. (1997). Generic metrics and connections on Spin- and Spinc-manifolds. Comm. Math. Phys. 188: 407–437

    Article  MATH  MathSciNet  Google Scholar 

  21. Mazzeo R.R., Melrose R.B. (1998). Pseudodifferential operators on manifolds with fibered boundaries. Asian J. Math. 2: 833–866

    MATH  MathSciNet  Google Scholar 

  22. Maniccia L., Panarese P. (2002). Eigenvalue asymptotics for a class of md-elliptic ψdo’s on manifolds with cylindrical exits. Ann. Mat. Pura Appl. (4) 181: 283–308

    Article  MATH  MathSciNet  Google Scholar 

  23. Melrose, R.B.: The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics 4, A. K. Peters, Wellesley (1993)

  24. Melrose R.B. (1995). The eta invariant and families of pseudodifferential operators. Math. Res. Lett. 2: 541–561

    MATH  MathSciNet  Google Scholar 

  25. Melrose, R.B., Nistor, V.: Homology of pseudodifferential operators I. Manifolds with boundary, funct-an/9606005

  26. Minakshisundaram S., Pleijel A. (1949). Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Can. J. Math. 1: 242–256

    MATH  MathSciNet  Google Scholar 

  27. Moroianu S. (2004). Adiabatic limits of eta and zeta functions of elliptic operators. Math. Z. 246: 441–471

    Article  MATH  MathSciNet  Google Scholar 

  28. Seeley, R.: Complex powers of an elliptic operator, Proc. Sympos. Pure Math. (Singular integrals), Chicago, 1966, Am. Math. Soc., pp. 288–307 (1967)

Download references

Author information

Authors and Affiliations

  1. IMAR, P.O. Box 1-764, RO-014700, Bucharest, Romania

    Sergiu Moroianu

  2. Şcoala Normală Superioară Bucharest, Calea Griviţei 21, Sector 1, Bucharest, Romania

    Sergiu Moroianu

Authors
  1. Sergiu Moroianu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergiu Moroianu.

Additional information

The author has been partially supported by the Research and Training Network HPRN-CT-1999-00118 “Geometric Analysis” funded by the European Commission.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moroianu, S. Weyl laws on open manifolds. Math. Ann. 340, 1–21 (2008). https://doi.org/10.1007/s00208-007-0137-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-007-0137-8

Mathematics Subject Classification (2000)

Search

Navigation