# Ngô Quốc Anh

## March 3, 2011

### The Paneitz operator in any dimension

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 16:43

Let us recall from this topic the following fact: Let $(M,g)$ be a compact Riemannian $4$-manifold, and let ${\rm Ric}_g$ and $R_g$ denote the Ricci tensor and the scalar curvature of $g$, respectively. The so-called Paneitz operator $P_g$ acts on a smooth function $u$ on $M$ via

$\displaystyle {P_g^4}(u) = \Delta _g^2u + {\rm div}\left( {\frac{2}{3}{R_g} - 2{\rm Ric}_g} \right)du$

which plays a similar role as the Laplace operator in dimension two where $d$ is the de Rham differential. Associated to this operator is the notion of $Q$-curvature given by

$\displaystyle Q_g^4=-\frac{1}{6}(\Delta R_g - R_g^2 +3|{\rm Ric}_g|_g^2).$

Under the following conformal change

$\widetilde g = e^{2u}g$

passing from $Q_g^4$ to $Q_{\widetilde g}^4$ is easy through the following formula

$P_g^4 (u)+Q_g^4=Q_{\widetilde g}^4e^{4u}.$

The above derivation is now well-known as the Paneitz operator and its $Q$-curvature done by Paneitz in his preprint in 1983 [here].

For dimension $n\geqslant 5$, this generalization is due to Branson [here]. Let $(M,g)$ be a Riemannian manifold of dimension $n \geqslant 5$ and define the operator

$P_g^n : C^4(M) \to C^0(M)$

by

$\displaystyle P_g^n=\Delta_g^2u-{\rm div}\left( a_n {R_g} +b_n {\rm Ric}_g\right)du + \frac{n-4}{2}Q_g^nu$

where

$\displaystyle a_n=\frac{(n-2)^2+4}{2(n-1)(n-2)}, \quad b_n=-\frac{4}{n-2}$

and

$\displaystyle Q_g^n=\frac{1}{2(n-1)}\Delta_gR_g+\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}R_g^2-\frac{2}{(n-2)^2}|{\rm Ric}_g|_g^2$

is the $Q$-curvature in dimension $n \geqslant 5$. Now under the following conformal change

$\displaystyle\widetilde g = u^\frac{4}{n-4}g$

passing from $Q_g^n$ to $Q_{\widetilde g}^n$ is easy through the following formula

$\displaystyle P_g^n (u)=\frac{n-4}{2}Q_{\widetilde g}^nu^\frac{n+4}{n-4}.$

In fact, this generalization also holds for $n=3$.

1. well done!

Comment by Fab — March 4, 2011 @ 15:39

2. Just for sake of completeness there is also a higher order version of Paneitz-Branson operator, called GJMS operator after this:
Graham, C. Robin; Jenne, Ralph; Mason, Lionel J.; Sparling, George A. J. (1992), “Conformally invariant powers of the Laplacian. I. Existence”, Journal of the London Mathematical Society. Second Series 46 (3): 557–565.
Fab

Comment by Fab — April 3, 2012 @ 22:40

• Thanks Fab for the reference.

Comment by Ngô Quốc Anh — April 3, 2012 @ 22:40