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


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).


Manifold Riemannian Manifold Zeta Function Dirac Operator Space Form 
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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

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