In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n. It is named after Stephen Paneitz, who discovered it in 1983, and whose preprint was later published posthumously in Paneitz 2008.
In fact, the same operator was found earlier in the context of conformal supergravity by E. Fradkin and A. Tseytlin in 1982 (Phys Lett B 110 (1982) 117 and Nucl Phys B 1982 (1982) 157 ).
Concretely, using the canonical trivialization of the density bundles in the presence of a metric, the Paneitz operator P can be represented in terms of a representative the Riemannian metric g as an ordinary operator on functions that transforms according under a conformal change g ↦ Ω2g according to the rule The operator was originally derived by working out specifically the lower-order correction terms in order to ensure conformal invariance.
The Paneitz operator has been most thoroughly studied in dimension four where it appears naturally in connection with extremal problems for the functional determinant of the Laplacian (via the Polyakov formula; see Branson & Ørsted 1991).
In dimension four only, the Paneitz operator is the "critical" GJMS operator, meaning that there is a residual scalar piece (the Q curvature) that can only be recovered by asymptotic analysis.
The Paneitz operator appears in extremal problems for the Moser–Trudinger inequality in dimension four as well (Chang 1999) There is a close connection between 4 dimensional Conformal Geometry and 3 dimensional CR geometry associated with the study of CR manifolds.
There is a naturally defined fourth order operator on CR manifolds introduced by C. Robin Graham and John Lee that has many properties similar to the classical Paneitz operator defined on 4 dimensional Riemannian manifolds.
The operator defined by Graham and Lee though defined on all odd dimensional CR manifolds, is not known to be conformally covariant in real dimension 5 and higher.
The conformal covariance of this operator has been established in real dimension 3 by Kengo Hirachi.
Here unlike changing the metric by a conformal factor as in the Riemannian case discussed above, one changes the contact form on the CR 3 manifold by a conformal factor.
This follows by the conformal covariant properties of the CR Paneitz operator first observed by Kengo Hirachi.
[2] Furthermore, the CR Paneitz operator plays an important role in obtaining the sharp eigenvalue lower bound for Kohn's Laplacian.
This is a result of Sagun Chanillo, Hung-Lin Chiu and Paul C.
[3] This sharp eigenvalue lower bound is the exact analog in CR Geometry of the famous André Lichnerowicz lower bound for the Laplace–Beltrami operator on compact Riemannian manifolds.
It allows one to globally embed, compact, strictly pseudoconvex, abstract CR manifolds into
There is also a partial converse of the above result where the authors, J. S. Case, S. Chanillo, P. Yang, obtain conditions that guarantee when embedded, compact CR manifolds have non-negative CR Paneitz operators.
[4] The formal definition of the CR Paneitz operator
denotes the Kohn Laplacian which plays a fundamental role in CR Geometry and several complex variables and was introduced by Joseph J. Kohn.
Accounts of the Webster-Tanaka, connection, Torsion and curvature tensor may be found in articles by John M. Lee and Sidney M.
[5][6] There is yet another way to view the CR Paneitz operator in dimension 3.
is the dual 1-form to the CR-holomorphic tangent vector that defines the CR structure on the compact manifold.
is now seen to be related to the CR Paneitz operator for the contact form
by the formula of Hirachi: Next note the volume forms on the manifold
satisfy Using the transformation formula of Hirachi, it follows that, Thus we easily conclude that: is a CR invariant.
That is the integral displayed above has the same value for different contact forms describing the same CR structure
where the Webster-Tanaka torsion tensor is zero, it is seen from the formula displayed above that only the leading terms involving the Kohn Laplacian survives.
Next from the tensor commutation formulae given in [5], one can easily check that the operators
This CR structure on ellipsoids has non-vanishing Webster-Tanaka torsion.
Results on when the kernel is exactly the pluriharmonic functions, the nature and role of the supplementary space in the kernel etc., may be found in the article cited as [4] below.
One of the principal applications of the CR Paneitz operator and the results in [3] are to the CR analog of the Positive Mass theorem due to Jih-Hsin Cheng, Andrea Malchiodi and Paul C.