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

By S. M. Paneitz. 2008

426 citations recorded by the source corpus.

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 It is surely one of the most cited unpublished articles.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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3f9337252e9f798665bb538f9370272dad39fe87

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

What is Q-Curvature? (2008, neutral): “This is because Pf is a divergence for any f .”
Fractional Laplacian in conformal geometry (2010, neutral): “[25]) P2 = (−∆)(2) + δ (anRg + bnRic) d+ n−4 2 Q2.”
Higher spins in AdS5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT (2014, neutral): “Then Õ is the 4-derivative Weyl invariant scalar operator of [27, 28].”
Fractional conformal Laplacians and fractional Yamabe problems (2010, background): “The conformal fractional Laplacians and fractional scalar curvatures should also be compared to the higher order generalization of the conformal Laplacian and scalar curvature: the Paneitz operator P ĥ 2 and its associated Q-curvature (see [38], [5], [39]).”
Local uniqueness and periodicity induced by concentration (2017, neutral): “For instance, when m = 2, problem (P) is related to the Paneitz operator, which was introduced by Paneitz [28] for smooth four-dimensional Riemannian manifolds and was generalized by Branson [6] to smooth N -dimensional Riemannian manifolds.”
Q‐Curvature on a Class of Manifolds with Dimension at Least 5 (2014, background): “1 Introduction Recall the definition of the fourth-order Paneitz operator and its associated Qcurvature [1, 27]: when .”
Constant Q-curvature metrics in arbitrary dimension (2007, background): “This operator discovered by Paneitz in 1983 ( see [ 46 ]) and the corresponding Q-curvature introduced by Branson ( see [9]) are defined in terms of Ricci tensor Ricg and scalar curvature Rg of the manifold (M,g) as follows”