Sharp inequalities, the functional determinant, and the complementary series

  title={Sharp inequalities, the functional determinant, and the complementary series},
  author={T. Branson},
  journal={Transactions of the American Mathematical Society},
  • T. Branson
  • Published 1995
  • Mathematics
  • Transactions of the American Mathematical Society
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 conformai change of metric. The… Expand

Tables from this paper

Sharp inequalities for functional integrals and traces of conformally invariant operators
The intertwining operators A d = Ad(g) on the round sphere(Sn, g) are the conformal analogues of the power Laplacians 1d/2 on the flatRn. To each metricρg, conformally equivalent to g, we canExpand
Nonlinear Phenomena in the Spectral Theory of Geometric Linear Diierential Operators
The extremal problem for the functional determinant of a natural linear elliptic operator a on Riemannian manifold is studied. Viewing the determinant as a function of the Riemannian metric, weExpand
Conformally invariant powers of the Laplacian — A complete nonexistence theorem
Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observationExpand
Shift operators, residue families and degenerate Laplacians
We introduce new aspects in conformal geometry of some very natural second-order differential operators. These operators are termed shift operators. In the flat space, they are intertwining operatorsExpand
Conformally Invariant Operators, Differential Forms, Cohomology and a Generalisation of Q-Curvature
On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundleExpand
Rigidity of Conformal Functionals on Spheres
In this paper we investigate the nature of stationary points of functionals on the space of Riemannian metrics on a smooth compact manifold. Special cases are spectral invariants associated withExpand
Conformal symmetry breaking differential operators on differential forms
We study conformal symmetry breaking differential operators which map differential forms on R to differential forms on a codimension one subspace R. These operators are equivariant with respect toExpand
Dirac Operator and Spectral Geometry
This paper is devoted to mathematical and physical properties of the Dirac operator and spectral geometry. Spin-structures in Lorentzian and Riemannian manifolds, and the global theory of the DiracExpand
Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds
In this paper we study bounds for the first eigenvalue of the Paneitz operator P and its associated third-order boundary operator B (see (1.1) and (1.13) for a precise definitions) on four-manifolds.Expand
Conformal invariants and partial differential equations
Our goal is to study quantities in Riemannian geometry which remain invariant under the “conformal change of metrics”–that is, under changes of metrics which stretch the length of vectors butExpand


Leading terms in the heat invariants
Let D be a second-order differential operator with leading symbol given by the metric tensor on a compact Riemannian manifold. The asymptotics of the heat kernel based on D are given by homogeneous,Expand
The functional determinant of a four-dimensional boundary value problem
Working on four-dimensional manifolds with boundary, we consider, elliptic boundary value problems (A, B), A being the interior and B the boundary operator. These problems (A, B) should be valued inExpand
Estimates and extremals for zeta function determinants on four-manifolds
AbstractLetA 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 positiveExpand
Leading terms in the heat invariants for the Laplacians of the De Rham signature, and spin complexes.
Let D be a vector bundle-valued differential operator with positive definite leading symbol on a compact, Riemannian manifold. Asymptotic expansions of the kernel function and L 2 trace of the heatExpand
Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality
where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtainsExpand
Group representations arising from Lorentz conformal geometry
Abstract It is shown that there exist conformally covariant differential operators D 2 l , k of all even orders 2 l , on differential forms of all orders k , in the double cover n of the nExpand
Explicit functional determinants in four dimensions
4 2 2 ABSTRACT. Working on the four-sphere S , a flat four-torus, S x S2, or a compact hyperbolic space, with a metric which is an arbitrary positive function times the standard one, we give explicitExpand
Invariants of the heat equation
Let M be a compact Riemannian manifold without boundary and let P: C°°V-* C°°F be a self-adjoint elliptic differential operator with positive definite leading symbol. The asymptotics of the heatExpand
Invariants of locally conformally flat manifolds
Let Af be a locally conformally flat manifold with metric g. Choose a local coordinate system on M so g = e2hx dx o dx where dx o dx is the Euclidean standard metric. A polynomial P in theExpand
Harmonic analysis in vector bundles associated to the rotation and spin groups
Abstract Let V be any vector bundle over the sphere S n which is associated to the principal bundle of oriented orthonormal frames, or to that of spin frames. We give an explicit formula for theExpand