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The Hirzebruch Signature Theorem for Conical Metrics Read chapter Read first chapter

2016 | OriginalPaper | Chapter

Guide to Elliptic Boundary Value Problems for Dirac-Type Operators

Authors : Christian Bär, Werner Ballmann

Published in: Arbeitstagung Bonn 2013

Publisher: Springer International Publishing

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Abstract

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contain local elliptic boundary conditions in the sense of Lopatinski and Shapiro as well as the Atiyah–Patodi–Singer boundary conditions. We discuss boundary regularity of solutions and also spectral and index theory. The emphasis is on providing the reader with a working knowledge.

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previous chapter Depth and the Local Langlands Correspondence
next chapter Symplectic and Hyperkähler Implosion
Appendix
Available only for authorised users
Footnotes
1
Here \([D,f] = D \circ (f \cdot \mathop{\mathrm{id}}\nolimits _{E}) - (f \cdot \mathop{\mathrm{id}}\nolimits _{F}) \circ D\).
 
2
The η-invariant of A is defined as the value of the meromorphic extension of η(s) = ∑ λ ≠ 0sign(λ) | λ | −s at s = 0, see [APS]. Here the sum is taken over all nonzero eigenvalues of A taking multiplicities into account. Hence the η-invariant is a measure for the asymmetry of the spectrum.
 
3
If χ commutes with σ D (ν ♭ ), then B χ and B −χ are adjoint to each other.
 
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T. Ackermann, J. Tolksdorf, The generalized Lichnerowicz formula and analysis of Dirac operators. J. Reine Angew. Math. 471, 23–42 (1996)MathSciNetMATH
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C. Bär, W. Ballmann, Boundary value problems for elliptic differential operators of first order, in Survey in Differential Geometry, vol. 17, ed. by H.-D. Cao, S.-T. Yau (International Press, Somerville, 2012), pp. 1–78
[BBC]
W. Ballmann, J. Brüning, G. Carron, Regularity and index theory for Dirac-Schrödinger systems with Lipschitz coefficients. J. Math. Pures Appl. 89, 429–476 (2008)MathSciNetCrossRefMATH
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B. Booss–Bavnbek, K. Wojciechowski, Elliptic Boundary Problems for Dirac Operators (Birkhäuser, Boston, 1993)
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P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (Publish or Perish, Wilmington, 1984)MATH
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M. Gromov, H.B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984)MathSciNetMATH
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H.B. Lawson, M.-L. Michelsohn, Spin Geometry (Princeton University Press, Princeton, 1989)MATH
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R.S. Palais, Seminar on the Atiyah-Singer Index Theorem. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih, and R. Solovay (Princeton University Press, Princeton, 1965)
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J. Roe, Partitioning noncompact manifolds and the dual Toeplitz problem, in Operator Algebras and Applications, vol. 1. London Mathematical Society Lecture Note Series, vol. 135 (Cambridge University Press, Cambridge, 1988), pp. 187–228
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Metadata
Title
Guide to Elliptic Boundary Value Problems for Dirac-Type Operators
Authors
Christian Bär
Werner Ballmann
Copyright Year
2016
Book
Arbeitstagung Bonn 2013

Print ISBN: 978-3-319-43646-3

Electronic ISBN: 978-3-319-43648-7

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-43648-7_3

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