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

Stephen M. Paneitz

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 It is surely one of the most cited unpublished articles.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

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 It is surely one of the most cited unpublished articles.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

Paper Structure

This paper contains 2 theorems, 7 equations.

Key Result

Theorem 1

Let $M$ be an arbitrary smooth manifold of dimension $n>2$, and assume given two pseudo-Riemannian metrics $g_1$ and $g_2$ on $M$ related by $g_1=p^2g_2$ for a positive function $p$. Then for all scalar functions $\phi$, When $n=4$, and the kernel of this operator is independent of the metric $g$, and also invariant under the ordinary pointwise action of the conformal group $(M,g_1)$ (or equival

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2