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Sharp inequalities, the functional determinant, and the complementary series


Author: Thomas P. Branson
Journal: Trans. Amer. Math. Soc. 347 (1995), 3671-3742
MSC: Primary 58G26; Secondary 22E46, 53A30
DOI: https://doi.org/10.1090/S0002-9947-1995-1316845-2
MathSciNet review: 1316845
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Abstract: Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions $ 2$, $ 4$, and $ 6$ for the functional determinants of operators which are well behaved under conformal change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Ørsted; the method is sufficiently streamlined here that we are able to present the sixdimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on $ {S^2}$, and in the standard conformal classes on $ {S^4}$ and $ {S^6}$. The $ {S^2}$ results are due to Onofri, and the $ {S^4}$ results to Branson, Chang, and Yang; the $ {S^6}$ results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant differential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of $ {\text{S}}{{\text{O}}_0}(m + 1,1)$, and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on $ {S^6}$, we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding $ L_1^2({S^6})\hookrightarrow {L^3}({S^6})$ for section spaces of trace free symmetric two-tensors.


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  • [A] D. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385-398. MR 960950 (89i:46034)
  • [BFG] M. Beals, C. Fefferman, and R. Grossman, Strictly pseudoconvex domains in $ {\mathbb{C}^n}$, Bull. Amer. Math. Soc. 8 (1983), 125-322. MR 684898 (85a:32025)
  • [Bec] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. 138 (1993), 213-242. MR 1230930 (94m:58232)
  • [Bes] A. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, $ 3$. Folge, Band 10, Springer-Verlag, Berlin, 1987. MR 867684 (88f:53087)
  • [Bl] D. Bleecker, Determination of a Riemannian metric from the first variation of its spectrum, Amer. J. Math. 107 (1985), 815-831. MR 796904 (86k:58124)
  • [Bo] H. Boerner, Darstellungen von Gruppen, Springer-Verlag, Berlin, 1955. MR 0075211 (17:710b)
  • [Bra1] T. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293-345. MR 832360 (88a:58212)
  • [Bra2] -, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), 199-291. MR 904819 (90b:22016)
  • [Bra3] -, Harmonic analysis in vector bundles associated to the rotation and spin groups, J. Funct. Anal. 106 (1992), 314-328. MR 1165857 (93h:58157)
  • [BCY] T. Branson, S.-Y. A. Chang, and P. Yang, Estimates and extremals for zeta function determinants on four-manifolds, Comm. Math. Phys. 149 (1992), 241-262. MR 1186028 (93m:58116)
  • [BG1] T. Branson and P. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), 245-272. MR 1032631 (90m:58201)
  • [BG2] -, The functional determinant of a four-dimensional boundary value problem, Trans. Amer. Math. Soc. 344 (1994), 479-531. MR 1240945 (94k:58155)
  • [BGØ] T. Branson, P. Gilkey, and B. Ørsted, Leading terms in the heat invariants, Proc. Amer. Math. Soc. 109 (1990), 437-450. MR 1014642 (90k:58227)
  • [BGØ] -, Leading terms in the heat invariants for the Laplacians of the de Rham, signature, and spin complexes, Math. Scand. 66 (1990), 307-319. MR 1075147 (91k:58124)
  • [BGP] T. Branson, P. Gilkey, and J. Pohjanpelto, Invariants of locally conformally flat manifolds, Trans. Amer. Math. Soc. 347 (1995), 939-954. MR 1282884 (95j:53069)
  • [] T. Branson and B. Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), 261-293. MR 869104 (88b:58131)
  • [] -, Conformal geometry and global invariants, Differential Geom. Appl. 1 (1991), 279-308. MR 1244447 (94k:58154)
  • [] -, Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc. 113 (1991), 669-682. MR 1050018 (92b:58238)
  • [Bru] F. Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math. France 84 (1956), 97-205. MR 0084713 (18:907i)
  • [ES] M. Eastwood and M. Singer, A conformally invariant Maxwell gauge, Phys. Lett. 107A (1985), 73-74. MR 774899 (86j:83031)
  • [E1] J. Escobar, Sharp constant in a Sobolev trace inequality, Indiana Univ. Math. J. 37 (1988), 687-698. MR 962929 (90a:46071)
  • [E2] -, The Yamabe problem on manifolds with boundary, J. Differential Geom. 35 (1992), 21-84. MR 1152225 (93b:53030)
  • [F] H. D. Fegan, Conformally invariant first order differential operators, Quart. J. Math. Oxford 27 (1976), 371-378. MR 0482879 (58:2920)
  • [FG] H. D. Fegan and P. Gilkey, Invariants of the heat equation, Pacific J. Math. 117 (1985), 233-254. MR 779919 (86g:58130)
  • [G1] P. Gilkey, Smooth invariants of a Riemannian manifold, Adv. in Math. 28 (1978), 1-10. MR 0478171 (57:17660)
  • [G2] -, Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compositio Math. 38 (1979), 201-240. MR 528840 (80i:53020)
  • [G3] -, The spectral geometry of the higher order Laplacian, Duke Math. J. 47 (1980), 511-528. MR 587163 (82b:58097)
  • [G4] -, Invariance theory, the heat equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Wilmington, DE, 1984. MR 783634 (86j:58144)
  • [G5] -, Leading terms in asymptotics of the heat equation, Contemp. Math. 73 (1988), 79-85. MR 954631 (89h:58199)
  • [GJMS] C. R. Graham, R. Jenne, L. Mason, and G. Sparling, Conformally invariant powers of the Laplacian, I: existence, J. London. Math. Soc. 46 (1992), 557-565. MR 1190438 (94c:58226)
  • [Kl] F. Klein, Vorlesungen über Höhere Geometrie, Springer-Verlag, Berlin, 1926.
  • [Kn] A. Knapp, Representation theory of semisimple groups, Princeton Univ. Press, Princeton, NJ, 1986. MR 855239 (87j:22022)
  • [Ko] Y. Kosmann, Dérivées de Lie des spineurs, Ann. Mat. Pura Appl. (4) 91 (1972), 317-395. MR 0312413 (47:971)
  • [Lic] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9. MR 0156292 (27:6218)
  • [Lie] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. 118 (1983), 349-374. MR 717827 (86i:42010)
  • [M] V. Molčanov, Representations of pseudo-orthogonal groups associated with a cone, Math. USSR-Sb. 10 (1970), 333-347.
  • [Ob] Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), 247-258. MR 0303464 (46:2601)
  • [On] E. Onofri, On the positivity of the effective action in a theory of random surfaces, Comm. Math. Phys. 86 (1982), 321-326. MR 677001 (84j:58043)
  • [OPS1] B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148-211. MR 960228 (90d:58159)
  • [OPS2] -, Compact isospectral sets of surfaces, J. Funct. Anal. 80 (1988), 212-234. MR 960229 (90d:58160)
  • [P] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint, 1983.
  • [Sc] R. Schimming, Lineare Differentialoperatoren zweiter Ordnung mit metrischem Hauptteil und die Methode der Koinzidenzwerte in der Riemannschen Geometrie, Beiträge Anal. 15 (1981), 77-91. MR 614779 (82h:58051)
  • [SW] E. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163-196. MR 0223492 (36:6540)
  • [St] R. Strichartz, Linear algebra of curvature tensors and their covariant derivatives, Canad. J. Math. 40 (1988), 1105-1143. MR 973512 (90c:53048)
  • [V] D. Vogan, Unitary representations of reductive Lie groups, Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 1987. MR 908078 (89g:22024)
  • [Wa] G. Warner, Harmonic analysis on semi-simple Lie groups I, Springer-Verlag, Berlin, 1972. MR 0498999 (58:16979)
  • [] V. Wünsch, On conformally invariant differential operators, Math. Nachr. 129 (1986), 269-281. MR 864639 (88a:58207)

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DOI: https://doi.org/10.1090/S0002-9947-1995-1316845-2
Article copyright: © Copyright 1995 American Mathematical Society

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