Geometric heat kernel coefficient for APS-type boundary conditions

Gorm Salomonsen

Journées équations aux dérivées partielles (1998)

  • page 1-19
  • ISSN: 0752-0360

Abstract

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I present an alternative way of computing the index of a Dirac operator on a manifold with boundary and a special family of pseudodifferential boundary conditions. The local version of this index theorem contains a number of divergence terms in the interior, which are higher order heat kernel invariants. I will present a way of associating boundary terms to those divergence terms, which are rather local of nature.

How to cite

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Salomonsen, Gorm. "Geometric heat kernel coefficient for APS-type boundary conditions." Journées équations aux dérivées partielles (1998): 1-19. <http://eudml.org/doc/93353>.

@article{Salomonsen1998,
abstract = {I present an alternative way of computing the index of a Dirac operator on a manifold with boundary and a special family of pseudodifferential boundary conditions. The local version of this index theorem contains a number of divergence terms in the interior, which are higher order heat kernel invariants. I will present a way of associating boundary terms to those divergence terms, which are rather local of nature.},
author = {Salomonsen, Gorm},
journal = {Journées équations aux dérivées partielles},
keywords = {Atiyah-Patodi-Singer; index; Dirac operator; pseudodifferential boundary conditions; heat kernel invariants},
language = {eng},
pages = {1-19},
publisher = {Université de Nantes},
title = {Geometric heat kernel coefficient for APS-type boundary conditions},
url = {http://eudml.org/doc/93353},
year = {1998},
}

TY - JOUR
AU - Salomonsen, Gorm
TI - Geometric heat kernel coefficient for APS-type boundary conditions
JO - Journées équations aux dérivées partielles
PY - 1998
PB - Université de Nantes
SP - 1
EP - 19
AB - I present an alternative way of computing the index of a Dirac operator on a manifold with boundary and a special family of pseudodifferential boundary conditions. The local version of this index theorem contains a number of divergence terms in the interior, which are higher order heat kernel invariants. I will present a way of associating boundary terms to those divergence terms, which are rather local of nature.
LA - eng
KW - Atiyah-Patodi-Singer; index; Dirac operator; pseudodifferential boundary conditions; heat kernel invariants
UR - http://eudml.org/doc/93353
ER -

References

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  1. [APS] Atiyah, M.F., Patodi, V.K., Singer, I.M. : «Spectral Asymmetry and Riemannian Geometry I.», Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69. Zbl0297.58008MR53 #1655a
  2. [BGV] Berline, N., Getzler, E., Vergne, M. : «Heat Kernels and Dirac Operators», Second Edition, Springer-Verlag (1996). 
  3. [BW] Boos, B., Wojciechowski, K. : «Elliptic Boundary Problems for Dirac Operators», Birkhauser, Boston (1993). 
  4. [Ca1] Carron, G. : « Inégalités de Sobolev et L2-Cohomologie », Séminaire de Théorie Spectrale et Géométrie GRENOBLE 1994-1995 (171-176) Zbl0937.58500MR1715966
  5. [Ca2] Carron, G. : « Un Théorème de L'Indice Relativ. » Séminaire de théorie Spectrale et Géométrie GRENOBLE 1996-1997 (193-202) Zbl0919.58061
  6. [Ca3] Carron, G. : « Théorèmes de L'Indice sur les Variétés Non-Compactes » Preprint, April 1998. Zbl1014.58012
  7. [Ch] Chernoff, P.R. : «Essential Self-adjointness of Powers of Generators of Hyperbolic Equations.» J. Funct. Anal. 12 (1973), 401-414. Zbl0263.35066MR51 #6119
  8. [Gi] Gilkey, P. : «Invariance Theory, the Heat Equation and the Atiyah-Patodi-Singer Index Theorem», 2nd edition, CRC Press (1995) Zbl0856.58001MR98b:58156
  9. [Gr1] Grubb, G.«Heat Trace Expansions for Elliptic Systems with Pseudodifferential Boundary Conditions.» Preprint, University of Copenhagen, November 1997. 
  10. [Gr2] Grubb, G.«Functional Calculus of Pseudodifferential Boundary problems» Birkhäuser, Boston (1986). Zbl0622.35001MR88f:35002
  11. [GS] Grubb, G., Seeley, R. : «Weakly Parametric Pseudodifferential Operators and Atiyah-Patodi-Singer Boundary Problems.», Invent. Math. 121 (1995) 481-529. Zbl0851.58043MR96k:58216
  12. [Gy] Gray, A. : «Tubes.» Addison-Wesley, New York, 1990 Zbl0692.53001MR92d:53002
  13. [HMM] Hassel, A., Mazzeo, R., Melrose, R.B.«A Signature Formula for Manifolds with Corners of Codimension Two» Topology 36 (1997). Zbl0883.58038MR98c:58163
  14. [KN] Kobayashi, S., Nomizu, K. : «Foundations of Differential Geometry». Volume I, Interscience Publishers, 1963. Zbl0119.37502MR27 #2945
  15. [LM] Lawson, H. B., Michelsohn, M. L. : «Spin Geometry.», Princeton University Press, Princeton, New Jersey, 1989. Zbl0688.57001MR91g:53001
  16. [Mü] Müller, W. : «On the L2-Index of Dirac Operators on Manifolds with Corners of Codimension Two. I» J. Diff. Geo. 44 (1996) 97-177. Zbl0881.58071
  17. [Pe] Pedersen, G.K. : «Analysis Now», Springer-Verlag, Graduate Texts in Mathematics 118 (1989) Zbl0668.46002MR90f:46001
  18. [Sa1] Salomonsen, G. : «The Atiyah-Patodi-Singer Index Theorem for Manifolds with Corners I.» To appear. Available on http://styx.math.uni-bonn.de/~gorm/ 

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