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 ). It is given by the formula
where Δ is the Laplace–Beltrami operator, d is the exterior derivative, δ is its formal adjoint, V is the Schouten tensor, J is the trace of the Schouten tensor, and the dot denotes tensor contraction on either index. Here Q is the scalar invariant
which in four dimensions yields the Q-curvature.
The operator is especially important in conformal geometry, because in a suitable sense it depends only on the conformal structure. Another operator of this kind is the conformal Laplacian. But, whereas the conformal Laplacian is second-order, with leading symbol a multiple of the Laplace–Beltrami operator, the Paneitz operator is fourth-order, with leading symbol the square of the Laplace–Beltrami operator. The Paneitz operator is conformally invariant in the sense that it sends conformal densities of weight 2 − n/2 to conformal densities of weight −2 − n/2. 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. Subsequent investigations have situated the Paneitz operator into a hierarchy of analogous conformally invariant operators on densities: the GJMS operators.
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)
CR Paneitz Operator
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 Jack Lee that has many properties similar to the Paneitz operator defined above on 4 dimensional Riemannian manifolds. This operator in CR Geometry is called the CR Paneitz operator. 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. It is always a non-negative operator in real dimension 5 and higher. 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. Non-negativity of the CR Paneitz operator in dimension 3 is a CR invariant condition as proved below. This follows by the conformal covariant properties of the CR Paneitz operator first observed by Kengo Hirachi. 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. Yang. This sharp eigenvalue lower bound is the exact analog in CR Geometry of the famous Andre Lichnerowicz lower bound for the Laplace-Beltrami operator on compact Riemannian manifolds. It allows one to globally embed, compact, strictly pseuodconvex, abstract CR manifolds into
Here
Hirachi's covariant transformation formula for
Next note the volume forms on the manifold
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
The operator
where
Thus
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. Yang. This allows one to obtain results on the CR Yamabe problem.
More facts related to the role of the CR Paneitz operator in CR geometry can be obtained from the article CR manifold.