Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Explicit functional determinants in four dimensions

Authors: Thomas P. Branson and Bent Ørsted
Journal: Proc. Amer. Math. Soc. 113 (1991), 669-682
MSC: Primary 58G26; Secondary 47F05, 58E11, 58G11
MathSciNet review: 1050018
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Working on the four-sphere $ {S^4}$, a flat four-torus, $ {S^2} \times {S^2}$, or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicit formulas for the functional determinants of the conformal Laplacian (Yamabe operator) and the square of the Dirac operator, and discuss qualitative features of the resulting variational problems. Our analysis actually applies in the conformal class of any Riemannian, locally symmetric, Einstein metric on a compact $ 4$-manifold; and to any geometric differential operator which has positive definite leading symbol, and is a positive integral power of a conformally covariant operator.

References [Enhancements On Off] (What's this?)

  • [B1] T. Branson, Conformally covariant equations on differential forms, Comm. Partial Differential Equations 7 (1982), 392-431. MR 652815 (84g:58110)
  • [B2] -, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), 293-345. MR 832360 (88a:58212)
  • [B3] -, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), 199-291. MR 904819 (90b:22016)
  • [B4] -, Second-order conformal covariants, preprint.
  • [B5] -, Harmonic analysis in vector bundles associated to the rotation and spin groups, J. Funct. Anal. (to appear). MR 1165857 (93h:58157)
  • [BCY] T. Branson, S.-Y. A. Chang, and P. Yang, Estimates and extremal problems for the zeta function determinant on four-manifolds, preprint.
  • [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] -, Residues of the eta function for an operator of Dirac type, preprint.
  • [BØ1] T. Branson and B. Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), 261-293. MR 869104 (88b:58131)
  • [BØ2] -, Conformal deformation and the heat operator, Indiana Univ. Math. J. 37 (1988), 83-110. MR 942096 (89g:58214)
  • [BØ3] -, Conformal geometry and global invariants, Differential Geometry and Applications (to appear).
  • [BPY] R. Brooks, P. Perry, and P. Yang, Isospectral sets of conformally equivalent metrics, Duke Math. J. 58 (1989), 131-150. MR 1016417 (90i:58193)
  • [CY1] S.-Y. A. Chang and P. Yang, Compactness of isospectral conformal metrics on $ {S^3}$, Comment. Math. Helv. 64 (1989), 363-374. MR 998854 (90c:58181)
  • [CY2] -, Isospectral conformal metrics on $ 3$-manifolds, J. Amer. Math. Soc. 3 (1990), 117-145. MR 1015647 (91c:58140)
  • [ES] M. Eastwood and M. Singer, A conformally invariant Maxwell gauge, Phys. Lett. 107A (1985), 73-74. MR 774899 (86j:83031)
  • [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, Spectral geometry of a Riemannian manifold, J. Differential Geometry 10 (1975), 601-618. MR 0400315 (53:4150)
  • [G2] -, The spectral geometry of the higher order Laplacian, Duke Math. J. 47 (1980), 511-528. MR 587163 (82b:58097)
  • [G3] -, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Publish or Perish, Wilmington, Delaware, 1984. MR 783634 (86j:58144)
  • [H] S. W. Hawking, Zeta function regularization of path integrals in curved spacetime, Comm. Math. Phys. 55 (1977), 133-148. MR 0524257 (58:25823)
  • [K] Y. Kosmann, Sur les degrés conformes des opérateurs différentiels, C. R. Acad. Sci. Paris Ser. A 280 (1975), 229-232. MR 0391188 (52:12009)
  • [OPS] B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148-211. MR 960228 (90d:58159)
  • [Pa] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint.
  • [Polc] J. Polchinski, Evaluation of the one loop string path integral, Comm. Math. Phys. 104 (1986), 37-47. MR 834480 (87j:81223)
  • [Poly1] A. Polyakov, Quantum geometry of Bosonic strings, Phys. Lett. B 103 (1981), 207-210. MR 623209 (84h:81093a)
  • [Poly2] -, Quantum geometry of Fermionic strings, Phys. Lett. B 103 (1981), 211-213. MR 623210 (84h:81093b)
  • [Sc] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geometry 20 (1984), 479-495. MR 788292 (86i:58137)
  • [Se] R. Seeley, Complex powers of an elliptic operator, Proc. Sympos. Pure Math. 10 (1967), 288-307. MR 0237943 (38:6220)
  • [We1] W. Weisberger, Normalization of the path integral measure and the coupling constants for bosonic strings, Nuclear Physics B 284 (1987), 171-200. MR 879081 (88g:81130)
  • [We2] -, Conformal invariants for determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 112 (1987), 633-638. MR 910583 (89c:58135)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58G26, 47F05, 58E11, 58G11

Retrieve articles in all journals with MSC: 58G26, 47F05, 58E11, 58G11

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society