Explicit functional determinants in four dimensions
HTML articles powered by AMS MathViewer
- by Thomas P. Branson and Bent Ørsted
- Proc. Amer. Math. Soc. 113 (1991), 669-682
- DOI: https://doi.org/10.1090/S0002-9939-1991-1050018-8
- PDF | Request permission
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
- Thomas P. Branson, Conformally covariant equations on differential forms, Comm. Partial Differential Equations 7 (1982), no. 4, 393–431. MR 652815, DOI 10.1080/03605308208820228
- Thomas P. Branson, Differential operators canonically associated to a conformal structure, Math. Scand. 57 (1985), no. 2, 293–345. MR 832360, DOI 10.7146/math.scand.a-12120
- Thomas P. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), no. 2, 199–291. MR 904819, DOI 10.1016/0022-1236(87)90025-5 —, Second-order conformal covariants, preprint.
- Thomas P. Branson, Harmonic analysis in vector bundles associated to the rotation and spin groups, J. Funct. Anal. 106 (1992), no. 2, 314–328. MR 1165857, DOI 10.1016/0022-1236(92)90050-S T. Branson, S.-Y. A. Chang, and P. Yang, Estimates and extremal problems for the zeta function determinant on four-manifolds, preprint.
- Thomas P. Branson and Peter B. Gilkey, The asymptotics of the Laplacian on a manifold with boundary, Comm. Partial Differential Equations 15 (1990), no. 2, 245–272. MR 1032631, DOI 10.1080/03605309908820686 —, Residues of the eta function for an operator of Dirac type, preprint.
- Thomas P. Branson and Bent Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60 (1986), no. 3, 261–293. MR 869104
- Thomas P. Branson and Bent Ørsted, Conformal deformation and the heat operator, Indiana Univ. Math. J. 37 (1988), no. 1, 83–110. MR 942096, DOI 10.1512/iumj.1988.37.37004 —, Conformal geometry and global invariants, Differential Geometry and Applications (to appear).
- Robert Brooks, Peter Perry, and Paul Yang, Isospectral sets of conformally equivalent metrics, Duke Math. J. 58 (1989), no. 1, 131–150. MR 1016417, DOI 10.1215/S0012-7094-89-05808-0
- Sun-Yung A. Chang and Paul C. Yang, Compactness of isospectral conformal metrics on $S^3$, Comment. Math. Helv. 64 (1989), no. 3, 363–374. MR 998854, DOI 10.1007/BF02564682
- Sun-Yung A. Chang and Paul C.-P. Yang, Isospectral conformal metrics on $3$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 1, 117–145. MR 1015647, DOI 10.1090/S0894-0347-1990-1015647-3
- Michael Eastwood and Michael Singer, A conformally invariant Maxwell gauge, Phys. Lett. A 107 (1985), no. 2, 73–74. MR 774899, DOI 10.1016/0375-9601(85)90198-7
- Howard D. Fegan and Peter Gilkey, Invariants of the heat equation, Pacific J. Math. 117 (1985), no. 2, 233–254. MR 779919
- Peter B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differential Geometry 10 (1975), no. 4, 601–618. MR 400315
- Peter B. Gilkey, The spectral geometry of the higher order Laplacian, Duke Math. J. 47 (1980), no. 3, 511–528. MR 587163
- Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
- S. W. Hawking, Zeta function regularization of path integrals in curved spacetime, Comm. Math. Phys. 55 (1977), no. 2, 133–148. MR 524257
- Yvette Kosmann, Sur les degrés conformes des opérateurs différentiels, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no. 4, Aiii, A229–A232 (French, with English summary). MR 391188
- B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no. 1, 148–211. MR 960228, DOI 10.1016/0022-1236(88)90070-5 S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint.
- Joseph Polchinski, Evaluation of the one loop string path integral, Comm. Math. Phys. 104 (1986), no. 1, 37–47. MR 834480
- A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), no. 3, 207–210. MR 623209, DOI 10.1016/0370-2693(81)90743-7
- A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), no. 3, 207–210. MR 623209, DOI 10.1016/0370-2693(81)90743-7
- Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
- R. T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) Amer. Math. Soc., Providence, R.I., 1967, pp. 288–307. MR 0237943
- William I. Weisberger, Normalization of the path integral measure and the coupling constants for bosonic strings, Nuclear Phys. B 284 (1987), no. 1, 171–200. MR 879081, DOI 10.1016/0550-3213(87)90032-0
- William I. Weisberger, Conformal invariants for determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 112 (1987), no. 4, 633–638. MR 910583
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 669-682
- MSC: Primary 58G26; Secondary 47F05, 58E11, 58G11
- DOI: https://doi.org/10.1090/S0002-9939-1991-1050018-8
- MathSciNet review: 1050018