Skip to main content
SpringerLink
Log in

Heat-trace asymptotics for edge Laplacians with algebraic boundary conditions

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We consider the Hodge-Laplace operator on manifolds with incomplete edge singularities and an intricate elliptic boundary value theory. We single out the class of algebraic self-adjoint extensions for the Hodge Laplacian. Our microlocal heat kernel construction for algebraic boundary conditions is guided by the method of signaling solutions by Mooers, though crucial arguments in the conical case obviously do not carry over to the setting of edges. We establish the heat kernel asymptotics for the algebraic extensions of the Hodge operator on edges, and elaborate on the exotic phenomena in the heat trace asymptotics which appear in the case of a non-Friedrichs extension.

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. M. Abramowitz and I. A. Stegun (eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1992.

    Google Scholar 

  2. P. Albin, A renormalized index theorem for some complete asymptotically regular metrics: the Gauss Bonnet theorem, Adv. Math. 213 (2007), 1–52.

    Article  MATH  MathSciNet  Google Scholar 

  3. E. Bahuaud, E. Dryden, and B. Vertman, Mapping properties of the heat operator on edge manifolds, Math. Nachr., to appear; preprint arXiv:1105.5119v2[math.AP].

  4. E. Bahuaud and B. Vertman, Yamabe flow on manifolds with edges, Math. Nachr. 287 (2014), 127–159.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Brüning and R. Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), 369–429.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Brüning and R. Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), 659–714.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983) 575–657 (1984).

    MATH  MathSciNet  Google Scholar 

  8. H. Falomir, M. A. Muschietti, P. A. G. Pisani, and R. Seeley, Unusual poles of the ξ-functions for some regular singular differential operators, J. Phys. A 36 (2003), 9991–10010.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. B. Gil, T. Krainer, and G. A. Mendoza, Trace expansions for elliptic cone operators with stationary domains, Trans. Amer. Math. Soc. 362 (2010), 6495–6522.

    Article  MATH  MathSciNet  Google Scholar 

  10. J. B. Gil, T. Krainer, and G. A. Mendoza, Dynamics on Grassmannians and resolvents of cone operators, Anal. PDE 4 (2011), 115–148.

    Article  MATH  MathSciNet  Google Scholar 

  11. K. Kirsten, P. Loya, and J. Park, The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d 2/dr 2 − 1/(4r 2), J. Math. Phys. 47 (2006), 043506.

    Article  MathSciNet  Google Scholar 

  12. K. Kirsten, P. Loya, and J. Park, Exotic expansions and pathological properties of ξ-functions on conic manifolds, with an Appendix by B. Vertman, J. Geom. Anal. 18 (2008), 835–888.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Lesch, Operators of Fuchs Type, Conical Singularities, and Asymptotic Methods, Stuttgart, 1997.

  14. M. Lesch and B. Vertma, Regular singular Sturm-Liouville operators and their zeta-determinants, J. Funct. Anal. 261 (2011), 408–450.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Mazzeo, Elliptic theory of differential edge operators. I. Comm. Partial Differential Equations 16 (1991), 1615–1664.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Mazzeo and B. Vertman, Analytic torsion on manifolds with edges, Adv. Math. 231 (2012), 1000–1040.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Melrose, Calculus of conormal distributions on manifolds with corners Internat. Math. Res. Notices 1992, 51–61.

  18. R. Melrose, The Atiyah-Patodi-Singer Index Theorem, A K Peters Ltd., Wellesley, MA, 1993.

    MATH  Google Scholar 

  19. E. Mooers, The heat kernel for manifolds with conic singularities, Ph.D. thesis at MIT, 1996. Available at http://dspace.mit.edu/bitstream/handle/1721.1/38406/36023220.pdf.

  20. E. Mooers, Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math. 78 (1999), 1–36.

    Article  MATH  MathSciNet  Google Scholar 

  21. B.W. Schulze, Pseudo-differential Operators on Manifolds with Singularities, North-Holland, Amsterdam, 1991.

    MATH  Google Scholar 

  22. B. Vertman, Zeta determinants for regular-singular Laplace-type operators, J. Math. Phys. 50 (2009), 083515.

    Article  MathSciNet  Google Scholar 

  23. B. Vertman, The exotic heat-trace asymptotics of a regular-singular operator revisited, J. Math. Phys. 54 (2013), 063501.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, 53115, Bonn, Germany

    Boris Vertman

Authors
  1. Boris Vertman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Vertman.

Additional information

Supported by the Hausdorff Research Institute at the University of Bonn.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vertman, B. Heat-trace asymptotics for edge Laplacians with algebraic boundary conditions. JAMA 125, 285–318 (2015). https://doi.org/10.1007/s11854-015-0009-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-015-0009-1

Keywords

Search

Navigation