Advertisement

Communications in Mathematical Physics

, Volume 149, Issue 2, pp 241–262 | Cite as

Estimates and extremals for zeta function determinants on four-manifolds

  • Thomas P. Branson
  • Sun-Yung A. Chang
  • Paul C. Yang
Article

Abstract

LetA be a positive integral power of a natural, conformally covariant differential operator on tensor-spinors in a Riemannian manifold. Suppose thatA is formally self-adjoint and has positive definite leading symbol. For example,A could be the conformal Laplacian (Yamabe operator)L, or the square of the Dirac operator
. Within the conformal class\(\left\{ {g = e^{2w} g_0 |w \in C^\infty (M)} \right\}\) of an Einstein, locally symmetric “background” metricgo on a compact four-manifoldM, we use an exponential Sobolev inequality of Adams to show that bounds on the functional determinant ofA and the volume ofg imply bounds on theW2,2 norm of the conformal factorw, provided that a certain conformally invariant geometric constantk=k(M, g o A) is strictly less than 32π2. We show for the operatorsL and
that indeedk < 32π2 except when (M, g o ) is the standard sphere or a hyperbolic space form. On the sphere, a centering argument allows us to obtain a bound of the same type, despite the fact thatk is exactly equal to 32π2 in this case. Finally, we use an inequality of Beckner to show that in the conformal class of the standard four-sphere, the determinant ofL or of
is extremized exactly at the standard metric and its images under the conformal transformation groupO(5,1).

Keywords

Manifold Riemannian Manifold Zeta Function Dirac Operator Space Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ad] Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math.128, 385–398 (1988)Google Scholar
  2. [Au1] Aubin, T.: Fonction de Green et valeurs propres du Laplacien. J. Math. Pures Appl.53, 347–371 (1974)Google Scholar
  3. [Au2] Aubin, T.: Meilleures constantes dans le théorèm, d'inclusion de Sobolev et un théorèm de Fredholm non linéaire pour la transformation conforme de la courbure scalaire. J. Funct. Anal.32, 148–174 (1979)CrossRefGoogle Scholar
  4. [Bec1] Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Preprint, to appear in the Annals of Math.Google Scholar
  5. [Bec2] Beckner, W.: private communicationGoogle Scholar
  6. [Bes] Besse, A.: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 10, Berlin, Heidelberg, New York: Springer 1987Google Scholar
  7. [BØ1] Branson, T., Ørsted, B.: Conformal indices of Riemannian manifolds. Comp. Math.60, 261–293 (1986)Google Scholar
  8. [BØ2] Branson, T., Ørsted, B.: Conformal deformation and the heat operator. Indiana U. Math. J.37, 83–110 (1988)CrossRefGoogle Scholar
  9. [BØ3] Branson, T., Ørsted, B.: Explicit functional determinants in four dimensions. Proc. Am. Math. Soc.113, 669–682 (1991)Google Scholar
  10. [CY] Chang, S.-Y.A., Yang, P.: Compactness of isospectral conformal metrics on S3. Comment. Math. Helv.64, 363–374 (1989)Google Scholar
  11. [F] Fontana, L.: Sharp borderline estimates on spheres and compact Riemannian manifolds. Ph.D. dissertation, Washington University, 1991Google Scholar
  12. [G] Gilkey, P.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Wilmington, DE: Publish or Perish 1984Google Scholar
  13. [J] Jensen, G.: Homogeneous Einstein spaces of dimension four. J. Diff Geom.3, 309–349 (1969)Google Scholar
  14. [MS] McKean, H., Singer, I.: Curvature and the eigenvalues of the Laplacian. J. Diff. Geom.1, 43–69 (1967)Google Scholar
  15. [O'N] O'Neil, R.: Convolution operators andL(p,q) spaces. Duke Math. J.30, 129–142 (1963)CrossRefGoogle Scholar
  16. [On] Onofri, E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys.86, 321–326 (1982)CrossRefGoogle Scholar
  17. [OV] Onofri, E., Virasoro, M.: On a formulation of Polyakov's string theory with regular classical solutions. Nucl. Phys. B201, 159–175 (1982)CrossRefGoogle Scholar
  18. [OPS1] Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal.80, 148–211 (1988)CrossRefGoogle Scholar
  19. [OPS2] Osgood, B., Phillips, R., Sarnak, P.: Compact isospectral sets of surfaces. J. Funct. Anal.80, 212–234 (1988)CrossRefGoogle Scholar
  20. [P1] Polyakov, A.: Quantum geometry of Bosonic strings. Phys. Lett. B.103, 207–210 (1981)CrossRefGoogle Scholar
  21. [P2] Polyakov, A.: Quantum geometry of Fermionic strings. Phys. Lett. B103, 211–213 (1981)CrossRefGoogle Scholar
  22. [RS] Ray, D., Singer, I.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.7, 145–210 (1971)CrossRefGoogle Scholar
  23. [S] Seeley, R.: Complex powers of an elliptic operator. Proc. Symposia Pure Math.10, 288–307 (1967)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Thomas P. Branson
    • 1
  • Sun-Yung A. Chang
    • 2
  • Paul C. Yang
    • 3
  1. 1.Department of MathematicsThe University of IowaIowa CityUSA
  2. 2.Department of MathematicsUCLALos AngelesUSA
  3. 3.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations