Laplacian Operators and Q-curvature on Conformally Einstein Manifolds

@article{Gover2005LaplacianOA,
  title={Laplacian Operators and Q-curvature on Conformally Einstein Manifolds},
  author={A. Rod Gover},
  journal={Mathematische Annalen},
  year={2005},
  volume={336},
  pages={311-334}
}
  • A. Gover
  • Published 2 June 2005
  • Mathematics
  • Mathematische Annalen
A new definition of canonical conformal differential operators Pk (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. These act between density bundles and, more generally, between weighted tractor bundles of any rank. By construction these factor into a power of a fundamental Laplacian associated to Einstein metrics. There are natural conformal Laplacian operators on density bundles due to Graham–Jenne–Mason–Sparling… 

Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

For even dimensional conformal manifolds several new conformally invariant objects were found recently: invariant differential complexes related to, but distinct from, the de Rham complex (these are

Conformal boundary operators, T-curvatures, and conformal fractional Laplacians of odd order

We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and

Conformal Operators on Weighted Forms; Their Decomposition and Null Space on Einstein Manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms $${L^\ell_k}$$Lkℓ which may be considered as the generalisation to differential forms of the

Conformal Invariants from Nodal Sets. I. Negative Eigenvalues and Curvature Prescription

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe

On conformally covariant powers of the Laplacian

We propose and discuss recursive formulas for conformally covariant powers $P_{2N}$ of the Laplacian (GJMS-operators). For locally conformally flat metrics, these describe the non-constant part of

Boundary Operators Associated With the Sixth-Order GJMS Operator

We describe a set of conformally covariant boundary operators associated with the 6th-order Graham--Jenne--Mason--Sparling (GJMS) operator on a conformally invariant class of manifolds that

Asymptotic expansions and conformal covariance of the mass of conformal differential operators

We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of Riesz

Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an

Conformally covariant differential operators acting on spinor bundles and related conformal covariants

Conformal powers of the Dirac operator on semi Riemannian spin manifolds are investigated. We give a new proof of the existence of conformal odd powers of the Dirac operator on semi Riemannian spin

Boundary calculus for conformally compact manifolds

On conformally compact manifolds of arbitrary signature, we use conformal geometry to identify a natural (and very general) class of canonical boundary problems. It turns out that these encompass and
...

References

SHOWING 1-10 OF 27 REFERENCES

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 bundle

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 observation

The ambient obstruction tensor and the conformal deformation complex

We construct here a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon

Conformally Invariant Powers of the Laplacian, Q-Curvature, and Tractor Calculus

Abstract: We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential

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 explicit

Ambient metric construction of Q-curvature in conformal and CR geometries

We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformally

Sharp inequalities, the functional determinant, and the complementary series

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

On Conformal Geometry.

  • T. Y. Thomas
  • Computer Science
    Proceedings of the National Academy of Sciences of the United States of America
  • 1926
Four functionals on the space of normalized almost Hermitian metrics on almost complex manifolds are discussed and the Euler-Lagrange equations for all these functionals are computed – as a tool for characterizing these metrics.

Notes on conformal differential geometry

This survey paper presents lecture notes from a series of four lectures addressed to a wide audience and it offers an introduction to several topics in conformal differential geometry. In particular,