A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)

Abstract

This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key role in conformal differential geometry in dimension 4 and is now known as the Paneitz operator.

This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

Cited by in corpus (117)

• Higher spins in AdS_5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT
• Anomalies, Conformal Manifolds, and Spheres
• On partition function and Weyl anomaly of conformal higher spin fields
• Explicit formulas for GJMS-operators and $Q$-curvatures
• Classification of solutions to the higher order Liouville's equation on R^{2m}
• Thermal transport, geometry, and anomalies
• Free box^k Scalar Conformal Field Theory
• Concentration-compactness phenomena in the higher order Liouville's equation
• Sign of Green's function of Paneitz operators and the Q curvature
• RG flows in d dimensions, the dilaton effective action, and the a-theorem
• $C_T$ for Non-unitary CFTs in Higher Dimensions
• Weyl vs. Conformal
• Arbitrary spin conformal fields in (A)dS
• Polyakov formulas for GJMS operators from AdS/CFT
• Restricted Weyl invariance in four-dimensional curved spacetime
• On the conformal properties of topological terms in even dimensions
• Liouville Conformal Field Theories in Higher Dimensions
• Superconformal duality-invariant models and $\mathcal{N}=4$ SYM effective action
• Gravitational Vacuum Condensate Stars
• Scale invariance vs conformal invariance
• Massive gravity generalization of $T\overline{T}$ deformations
• On conformally covariant powers of the Laplacian
• Existence theorems of the fractional Yamabe problem
• A complete study of the lack of compactness and existence results of a Fractional Nirenberg Equation via a flatness hypothesis: Part I
• On the realization of impossible anomalies
• Asymptotic behavior of Palais-Smale sequences associated with fractional Yamabe type equations
• The Effective Theory of Gravity and Dynamical Vacuum Energy
• Asymptotic analysis for fourth order Paneitz equations with critical growth
• Sobolev-Trace inequalities of order four
• Entanglement Entropy in Generalised Quantum Lifshitz Models
• GJMS-type Operators on a compact Riemannian manifold: Best constants and Coron-type solutions
• Weyl and Ricci gauging from the coset construction
• Tensor instabilities at the end of the LCDM universe
• Iterating free-field AdS/CFT: higher spin partition function relations
• Conformal Metrics with Constant Q-Curvature
• Compactness of conformal metrics with constant $Q$-curvature. I
• Nonlocal action for the super-Weyl anomalies: A new representation
• CP-violating super Weyl anomaly
• Exact quantum conformal symmetry, its spontaneous breakdown, and gravitational Weyl anomaly
• Second Eigenvalue of Paneitz Operators and Mean Curvature
• Conformal coupling of the scalar field with gravity in higher dimensions and invariant powers of the Laplacian
• On Wald entropy of black holes: logarithmic corrections and trace anomaly
• Nonuniqueness of conformal metrics with constant $Q$-curvature
• Integration of trace anomaly in 6D
• On the structure of the conformal higher-spin wave operators
• Conformal invariant powers of the Laplacian, Fefferman-Graham ambient metric and Ricci gauging
• Anomalous Gravitational TTT Vertex, Temperature Inhomogeneity, and Pressure Anisotropy
• Non-compact duality, super-Weyl invariance and effective actions
• Notes on conformal anomaly, nonlocal effective action and the metamorphosis of the running scale
• $\mathcal{N}=1$ Liouville SCFT in Four Dimensions
• Lectures on the fourth order Q curvature equation
• A symmetric 2-tensor canonically associated to Q-curvature and its applications
• One-loop renormalization of Lorentz/CPT violating scalar field theory in curved space-time
• Holographic Trace Anomaly and Local Renormalization Group
• $Q$-curvature and gravity
• The higher-dimensional Chern-Gauss-Bonnet formula for singular conformally flat manifolds
• Q curvature on a class of manifolds with dimension at least 5
• Trace anomaly and induced action for a metric-scalar background
• The sharp estimates for the first eigenvalue of Paneitz operator on 4-dimensional submanifolds
• Existence and local uniqueness of bubbling solutions for poly-harmonic equations with critical growth
• On constant Q-curvature metrics with isolated singularities
• On higher spin partition functions
• Nonlocal Quantum Effective Actions in Weyl-Flat Spacetimes
• Q-curvature Flow for GJMS Operators with Non-trivial Kernel
• On the vector conformal models in an arbitrary dimension
• Quantum Cosmology in Four Dimensions
• A different kind of four dimensional brane for string theory
• Constructing the independent basis of total derivative curvature-dependent terms in $6D$
• Logarithmic singularities of Schwartz kernels and local invariants of conformal and CR structures
• Liouville Conformal Field Theory on even-dimensional spheres
• Q-Curvature, Spectral Invariants, and Representation Theory
• Conformal geometry of the supercotangent and spinor bundles
• Constant Q-curvature metrics with Delaunay ends: the nondegenerate case
• Deformations of Q-curvature II
• Integral pinched 3-manifolds are space forms
• Conformal Geometry on Four Manifolds
• On a fourth order conformal invariant
• Asymptotic Expansions and Conformal Covariance of the Mass of Conformal Differential Operators
• Weyl invariance, non-compact duality and conformal higher-derivative sigma models
• Existence results for the fractional Nirenberg problem
• A perturbation result for the $Q_{gamma}$ curvature problem on ${\mathbb{S}}^n$
• Extremal metrics for the Paneitz Operator on closed four-Manifolds
• Conformal invariants from nodal sets. I. Negative Eigenvalues and Curvature Prescription
• Fractional Laplacian in Conformal Geometry
• Some Conformal Invariants from the Noncommutative Residue for Manifolds with Boundary
• Curvature flows on four manifolds with boundary
• Compactness of conformally compact Einstein 4-manifolds II
• On the Obata Theorem in a weighted Sasakian manifold
• Conformal symmetry breaking on differential forms and some applications
• Energy in Fourth Order Gravity
• Local(ish) gravity theories in conformal superspace
• Local estimates for conformal $Q$-curvature equations
• A Liouville type theorem of the linearly perturbed Paneitz equation on $S^3$
• Global existence and convergence for the CR Q-curvature flow in a closed strictly pseudoconvex CR 3-manifold
• Conformally Covariant Operators and Conformal Invariants on Weighted Graphs
• Riemannian manifolds with positive Yamabe invariant and Paneitz operator
• Paneitz operator for metrics near $S^3$
• Non-compactness of the Prescribed Q-curvature Problem in Large Dimensions
• Large singular solutions for conformal $Q$-curvature equations on ${\mathbb{S}}^n$
• The cosmology of a fundamental scalar
• Estimates for eigenvalues of the Paneitz operator
• Variational theory for the resonant T-curvature equation
• On Green's functions for positive, self-adjoint, elliptic pseudo-differential operators on closed, Riemannian manifolds
• Green function rigidity for two dimensional sphere
• Anomaly-induced vacuum effective action with torsion: covariant solution and ambiguities
• Sharp k-order Sobolev inequalities in the hyperbolic space ${\mathbb{H}}^n$
• Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity
• On a Fourth Order Lichnerowicz Type Equation Involving The Paneitz-Branson Operator
• Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension
• Limit of Fractional Power Sobolev Inequalities
• On the second Paneitz-Branson invariant
• A fractional conformal curvature flow on the unit sphere
• On the principal eigenfunction of positive elliptic differential operators and the prescription of $Q$-curvature on closed Riemannian manifolds
• Integral pinching results for manifolds with boundary
• Connected sums of closed Riemannian manifolds and fourth order conformal invariants
• Singular limits in higher order Lioville-type equations
• Bounce and stability in the early cosmology with anomaly-induced corrections