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Cauchy Kernels for some Conformally Flat Manifolds

  • John Ryan
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Here we will consider examples of conformally flat manifolds that are conformally equivalent to open subsets of the sphere S n . For such manifolds we shall introduce a Cauchy kernel, Cauchy integral formula for sections taking values in a spinor bundle and annihilated by a Dirac operator, or generalized Cauchy-Riemann operator. Basic properties of this kernel are examined. We also introduce a Green’s kernel and a Green’s formula for harmonic sections in this context.

Keywords

Conformally flat manifolds surgery Cauchy kernels Dirac operator 

Mathematics Subject Classification (2000)

30G35, 42B30, 53C27, 58J32 

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References

  1. [1]
    L. Ahlfors, Möbius transformations expressed through 2 x 2 matrices of Clifford numbers, Complex Variables 5 (1986), 215–224.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, Springer-Verlag, Heidelberg, 1992.CrossRefzbMATHGoogle Scholar
  3. [3]
    F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis,Pitman, London, 1982.zbMATHGoogle Scholar
  4. [4]
    T. Branson, G. Olafsson and P. GilkeyInvariants of conformally flat manifolds, Transactions of the A. M. S. 347 (1995), 939–954.CrossRefzbMATHGoogle Scholar
  5. [5]
    D. Calderbank, Dirac operators and Clifford analysis on manifolds with boundary, Max Plank Institute of Mathematics, Bonn, preprint no 96–131, 1996.Google Scholar
  6. [6]
    A. Chang, J. Qing, P. Yang, Compactification of a class of conformally flat manifolds, Invent. Math. 142 (2000), 65–93.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Cnops, An Introduction to Dirac Operators on Manifolds, Progress in Mathematical Physics, Birkhaüser, Boston, 2002.CrossRefzbMATHGoogle Scholar
  8. [8]
    J. Cnops and H. Malonek, An introduction to Clifford analysis Textos de Matematica, Serie B, 7, Universidade de Coimbra, Departmento de Matematica, Coimbra, 1995.Google Scholar
  9. [9]
    J. Elstrodt, F. Grunewald and J. Mennicke, Vahlen’s group of Clifford matrices, Math. Z. 196 (1987), 369–390.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    W. Goldman and Y. Kamishima, Conformal automorphism and conformally flat manifolds,Transactions of the AMS 323 (1991), 797–810.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    R. S. Krausshar and J. Ryan, Clifford and harmonic analysis on spheres and tori, to appear in Revista Matematica Iberoamericana.Google Scholar
  12. [12]
    R. S. Krausshar and J. Ryan, Some conformally flat spin manifolds,Dirac operators and automorphic forms,to appear.Google Scholar
  13. [13]
    H. B. Lawson and M-L. Michelson, Spin Geometry, Princeton University Press, Princeton, 1989.zbMATHGoogle Scholar
  14. [14]
    C. Li, A. McIntosh and T. Qian, Clifford algebras, Fourier transformations, and singular convolution operators on Lipschitz surfaces, Revista Matematica Iberoamericana 10 (1994), 665–721.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    C. Li, A. McIntosh and S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. of the AMS. 5 (1992), 455–481.MathSciNetzbMATHGoogle Scholar
  16. [16]
    H. Liu and J. Ryan, Clifford analysis techniques for spherical pde, J. Fourier Anal. Appl. 8 (2002), 535–563.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    H. Liu and J. Ryan, The conformal Laplacian on spheres and hyperbolas via Clifford analysis, Clifford Analysis and its Applications, F. Brackx et al, editors, Kluwer, Dordrecht, 2001, 255–266.CrossRefGoogle Scholar
  18. [18]
    A. McIntosh, Clifford algebras, Fourier theory, singular integrals and harmonic functions on Lipschitz domains, Clifford Algebras in Analysis and Related Topics, edited by J. Ryan, CRC Press, Boca Raton, 1996, 33–87.Google Scholar
  19. [19]
    M. Mitrea, Generalized Dirac operators on non-smooth manifolds and Maxwell’s equations,J. Fourier Anal. Appl. 7 (2001), 207–256.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    J. D. Moore, Lectures on Seiberg-Witten Invariants, Lecture Notes in Mathematics, no 1629, Springer Verlag, Heidelberg, 2001.Google Scholar
  21. [21]
    J. Peetre and T. Qian, Möbius covariance of iterated Dirac operators, J. Australian Math. Soc., Ser. A 56 (1994), 403–414.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    I. Porteous, Clifford algebras and the Classical Groups, Cambridge University Press, Cambridge, 1995.CrossRefzbMATHGoogle Scholar
  23. [23]
    J. Ryan, Dirac operators on spheres and hyperbolae, Bolletin de la Sociedad Matematica a Mexicana 3 (1996), 255–270.Google Scholar
  24. [24]
    J. Ryan, Clifford analysis on spheres and hyperbolae, Mathematical Methods in the Applied Sciences 20 (1997), 1617–1624.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    R. Schoen and S-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature,Inventiones Mathematicae 92 (1988), 47–71.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    E. Stein, Singular Integrals and Differentiability Properties of Functions,Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar
  27. [27]
    K. Th. Vahlen, Ober Bewegungen und Complexe Zahlen, Math. Ann. 55 (1902), 585–593.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    P. Van Lancker, Clifford analysis on the sphere, Clifford Algebras and their Applications in Mathematical Physics, Aachen 1996, V. Dietrich et al, editors, Kluwer, Dordrecht, 1998, 201–215.Google Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • John Ryan
    • 1
  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA

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