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$Q$-Curvature, Spectral Invariants, and Representation Theory
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: conformal differential geometry; functional determinant; conformal index. 
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     author = {Thomas P. Branson},
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}
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Thomas P. Branson. $Q$-Curvature, Spectral Invariants, and Representation Theory. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a89/

[1] Aubin T., “Équations differentielles non linéares et problème de Yamabe concernant la courbure scalaire”, J. Math. Pures Appl. (9), 55 (1976), 269–296 | MR | Zbl

[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 | DOI | MR | DOI | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[5] Beckner W., “Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality”, Ann. of Math. (2), 138 (1993), 213–242 | DOI | MR | Zbl

[6] Branson T., “Differential operators canonically associated to a conformal structure”, Math. Scand., 57 (1985), 293–345 | MR | Zbl

[7] Branson T., “Group representations arising from Lorentz conformal geometry”, J. Funct. Anal., 74 (1987), 199–291 | DOI | MR | Zbl

[8] Branson T., The functional determinant, Lecture Note Series, 4, Global Analysis Research Center, Seoul National University, 1993 | MR

[9] Branson T., “Sharp inequalities, the functional determinant, and the complementary series”, Trans. Amer. Math. Soc., 347 (1995), 3671–3742 | DOI | MR | Zbl

[10] Branson T., “An anomaly associated to 4-dimensional quantum gravity”, Comm. Math. Phys., 178 (1996), 301–309 | DOI | MR | Zbl

[11] Branson T., “Stein–Weiss operators and ellipticity”, J. Funct. Anal., 151 (1997), 334–383 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[13] Branson T., Gilkey P., “The asymptotics of the Laplacian on a manifold with boundary”, Comm. Partial Differential Equations, 15 (1990), 245–272 | DOI | MR | Zbl

[14] Branson T., Gilkey P., Pohjanpelto J., “Invariants of conformally flat manifolds”, Trans. Amer. Math. Soc., 347 (1995), 939–954 | DOI | MR

[15] Branson T., Ørsted B., “Conformal indices of Riemannian manifolds”, Compos. Math., 60 (1986), 261–293 | MR | Zbl

[16] Branson T., Ørsted B.,, “Explicit functional determinants in four dimensions”, Proc. Amer. Math. Soc., 113 (1991), 669–682 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[19] Carlen E., Loss M., “Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on $S^n$”, Geom. Funct. Anal., 2 (1992), 90–104 | DOI | MR | Zbl

[20] Dowker J. S., Kennedy G., “Finite temperature and boundary effects in static space-times”, J. Phys. A: Math. Gen., 11 (1978), 895–920 | DOI | MR

[21] Eastwood M., Singer M., “A conformally invariant Maxwell gauge”, Phys. Lett. A, 107 (1985), 73–74 | DOI | MR | Zbl

[22] Eastwood M., Slovák J., “Semiholonomic Verma modules”, J. Algebra, 197 (1997), 424–448 | DOI | MR | Zbl

[23] Gilkey P., Invariance theory, the heat equation, and the Atiyah–Singer index theorem, CRC Press, Boca Raton, 1995 | MR | Zbl

[24] Graham C. R., “Conformally invariant powers of the Laplacian. II. Nonexistence”, J. London Math. Soc. (2), 46 (1992), 566–576 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[26] Knapp A., Stein E., “Intertwining operators for semisimple Lie groups. II”, Invent. Math., 60 (1980), 9–84 | DOI | MR | Zbl

[27] Onofri E., “On the positivity of the effective action in a theory of random surfaces”, Comm. Math. Phys., 86 (1982), 321–326 | DOI | MR | Zbl

[28] Palais R. S., Foundations of global non-linear analysis, Benjamin Co., New York, 1968 | MR | Zbl

[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 | MR | Zbl

[31] Penrose R., Rindler W., Spinors and space-time, Vol. I, Cambridge University Press, 1984 | MR | Zbl

[32] Riegert R., “A non-local action for the trace anomaly”, Phys. Lett. B, 134 (1984), 56–60 | DOI | MR | Zbl

[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 | MR | Zbl

[34] Schoen R., “Conformal deformation of a Riemannian metric to constant scalar curvature”, J. Differential Geom., 20 (1984), 479–495 | MR | Zbl

[35] Seeley R., “Complex powers of an elliptic operator”, Proc. Symposia Pure Math., 10 (1967), 288–307 | MR | Zbl

[36] Trudinger N., “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds”, Ann. Scuola Norm. Sup. Pisa (3), 3 (1968), 265–274 | MR | Zbl

[37] Weyl H., The classical groups: their invariants and representations, Princeton University Press, 1939 | MR

[38] Widom H., “Szegő's theorem and a complete symbolic calculus for pseudodifferential operators”, Seminar on Singularities, Ann. of Math. Stud., 91, ed. L. Hörmander, 1979, 261–283 | MR | Zbl

[39] Wünsch V., “On conformally invariant differential operators”, Math. Nachr., 129 (1986), 269–281 | DOI | MR | Zbl

[40] Yamabe H., “On a deformation of Riemannian structures on compact manifolds”, Osaka J. Math., 12 (1960), 21–37 | MR | Zbl