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.
@article{JEDP_1998____A11_0, author = {Salomonsen, Gorm}, title = {Geometric heat kernel coefficient for {APS-type} boundary conditions}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, eid = {11}, pages = {1--19}, publisher = {Universit\'e de Nantes}, year = {1998}, mrnumber = {99h:58180}, zbl = {01808720}, language = {en}, url = {http://archive.numdam.org/item/JEDP_1998____A11_0/} }
Salomonsen, Gorm. Geometric heat kernel coefficient for APS-type boundary conditions. Journées équations aux dérivées partielles (1998), article no. 11, 19 p. http://archive.numdam.org/item/JEDP_1998____A11_0/
[APS] «Spectral Asymmetry and Riemannian Geometry I.», Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69. | MR | Zbl
, , :[BGV] «Heat Kernels and Dirac Operators», Second Edition, Springer-Verlag (1996).
, , :[BW] «Elliptic Boundary Problems for Dirac Operators», Birkhauser, Boston (1993).
, :[Ca1] « Inégalités de Sobolev et L2-Cohomologie », Séminaire de Théorie Spectrale et Géométrie GRENOBLE 1994-1995 (171-176) | Numdam | MR | Zbl
:[Ca2] « Un Théorème de L'Indice Relativ. » Séminaire de théorie Spectrale et Géométrie GRENOBLE 1996-1997 (193-202) | Numdam | Zbl
:[Ca3] « Théorèmes de L'Indice sur les Variétés Non-Compactes » Preprint, April 1998. | Zbl
:[Ch] «Essential Self-adjointness of Powers of Generators of Hyperbolic Equations.» J. Funct. Anal. 12 (1973), 401-414. | MR | Zbl
:[Gi] «Invariance Theory, the Heat Equation and the Atiyah-Patodi-Singer Index Theorem», 2nd edition, CRC Press (1995) | MR | Zbl
:[Gr1] «Heat Trace Expansions for Elliptic Systems with Pseudodifferential Boundary Conditions.» Preprint, University of Copenhagen, November 1997.
[Gr2] «Functional Calculus of Pseudodifferential Boundary problems» Birkhäuser, Boston (1986). | MR | Zbl
[GS] «Weakly Parametric Pseudodifferential Operators and Atiyah-Patodi-Singer Boundary Problems.», Invent. Math. 121 (1995) 481-529. | MR | Zbl
, :[Gy] «Tubes.» Addison-Wesley, New York, 1990 | MR | Zbl
:[HMM] «A Signature Formula for Manifolds with Corners of Codimension Two» Topology 36 (1997). | MR | Zbl
, ,[KN] «Foundations of Differential Geometry». Volume I, Interscience Publishers, 1963. | MR | Zbl
, :[LM] «Spin Geometry.», Princeton University Press, Princeton, New Jersey, 1989. | MR | Zbl
, :[Mü] «On the L2-Index of Dirac Operators on Manifolds with Corners of Codimension Two. I» J. Diff. Geo. 44 (1996) 97-177. | Zbl
:[Pe] «Analysis Now», Springer-Verlag, Graduate Texts in Mathematics 118 (1989) | MR | Zbl
:[Sa1] «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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