Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 090, 31 pages      arXiv:0709.2471      https://doi.org/10.3842/SIGMA.2007.090
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Q-Curvature, Spectral Invariants, and Representation Theory

Thomas P. Branson
Deceased

Received August 01, 2007 from Xingwang Xu; Published online September 16, 2007

Abstract
We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on even-dimensional conformal manifolds. The exposition is self-contained, in the sense of giving references sufficient to allow the reader to work through all details.

Key words: conformal differential geometry; functional determinant; conformal index.

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References

  1. Aubin T., Équations differentielles non linéares et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), 269-296.
  2. Avramidi I.G., A covariant technique for the calculation of the one-loop effective action, Nuclear Phys. B 355 (1991), 712-754, Errata, Nuclear Phys. B 509 (1998), 557-558.
  3. Avramidi I.G., Branson T., Heat kernel asymptotics of operators with non-Laplace principal part, Rev. Math. Phys. 13 (2001), 847-890, math-ph/9905001.
  4. Avramidi I.G., Branson T., A discrete leading symbol and spectral asymptotics for natural differential operators, J. Funct. Anal. 190 (2002), 292-337, hep-th/0109181.
  5. Beckner W., Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2) 138 (1993), 213-242.
  6. Branson T., Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293-345.
  7. Branson T., Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), 199-291.
  8. Branson T., The functional determinant, Lecture Note Series, Vol. 4, Global Analysis Research Center, Seoul National University, 1993.
  9. Branson T., Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995), 3671-3742.
  10. Branson T., An anomaly associated to 4-dimensional quantum gravity, Comm. Math. Phys. 178 (1996), 301-309.
  11. Branson T., Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997), 334-383.
  12. Branson T., Chang S.-Y.A., Yang P., Estimates and extremals for zeta function determinants on four-manifolds, Comm. Math. Phys. 149 (1992), 241-262.
  13. Branson T., Gilkey P., The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), 245-272.
  14. Branson T., Gilkey P., Pohjanpelto J., Invariants of conformally flat manifolds, Trans. Amer. Math. Soc. 347 (1995), 939-954.
  15. Branson T., Ørsted B., Conformal indices of Riemannian manifolds, Compos. Math. 60 (1986), 261-293.
  16. Branson T., Ørsted B., Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc. 113 (1991), 669-682.
  17. Branson T., Peterson L.J., in preparation.
  18. Browder F.E., Families of linear operators depending on a parameter, Amer. J. Math. 87 (1965), 752-758.
  19. Carlen E., Loss M., Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on Sn, Geom. Funct. Anal. 2 (1992), 90-104.
  20. Dowker J.S., Kennedy G., Finite temperature and boundary effects in static space-times, J. Phys. A: Math. Gen. 11 (1978), 895-920.
  21. Eastwood M., Singer M., A conformally invariant Maxwell gauge, Phys. Lett. A 107 (1985), 73-74.
  22. Eastwood M., Slovák J., Semiholonomic Verma modules, J. Algebra 197 (1997), 424-448.
  23. Gilkey P., Invariance theory, the heat equation, and the Atiyah-Singer index theorem, CRC Press, Boca Raton, 1995.
  24. Graham C.R., Conformally invariant powers of the Laplacian, II. Nonexistence, J. London Math. Soc. (2) 46 (1992), 566-576.
  25. Graham C.R., Jenne R., Mason L., Sparling G., Conformally invariant powers of the Laplacian, I. Existence, J. London Math. Soc. (2) 46 (1992), 557-565.
  26. Knapp A., Stein E., Intertwining operators for semisimple Lie groups. II, Invent. Math. 60 (1980), 9-84.
  27. Onofri E., On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys. 86 (1982), 321-326.
  28. Palais R.S., Foundations of global non-linear analysis, Benjamin & Co., New York, 1968.
  29. Paneitz S., A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, 1983.
  30. Parker T., Rosenberg S., Invariants of conformal Laplacians, J. Differential Geom. 25 (1987), 199-222.
  31. Penrose R., Rindler W., Spinors and space-time, Vol. I, Cambridge University Press, 1984.
  32. Riegert R., A non-local action for the trace anomaly, Phys. Lett. B 134 (1984), 56-60.
  33. Schimming R., Lineare Differentialoperatoren zweiter Ordnung mit metrischem Hauptteil und die Methode der Koinzidenzwerte in der Riemannschen Geometrie, Beitr. z. Analysis 15 (1981), 77-91.
  34. Schoen R., Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), 479-495.
  35. Seeley R., Complex powers of an elliptic operator, Proc. Symposia Pure Math. 10 (1967), 288-307.
  36. Trudinger N., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 3 (1968), 265-274.
  37. Weyl H., The classical groups: their invariants and representations, Princeton University Press, 1939.
  38. Widom H., Szegö's theorem and a complete symbolic calculus for pseudodifferential operators, in Seminar on Singularities, Editor L. Hörmander, Ann. of Math. Stud. 91 (1979), 261-283.
  39. Wünsch V., On conformally invariant differential operators, Math. Nachr. 129 (1986), 269-281.
  40. Yamabe H., On a deformation of Riemannian structures on compact manifolds, Osaka J. Math. 12 (1960), 21-37.


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