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.

3 Comments »

  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


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