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}.

(more…)

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January 27, 2011

Prescribed Q-curvature on 4-manifolds

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 3:40

Let (M,g) be a compact Riemannian 4-manifold, and let {\rm Ric} and R denote the Ricci tensor and the scalar curvature of g, respectively.

A natural conformally invariant in dimension four is

\displaystyle Q=Q_g=-\frac{1}{6}(\Delta R - R^2 +3|{\rm Ric}|^2).

This Q is commonly refered to the Q-curvature of metric g. The term

R^2 -3|{\rm Ric}|^2

is commonly denoted by 6\sigma_2(A) where

\displaystyle A={\rm Ric}-\frac{1}{6}Rg

the Schouten tensor of g and

\displaystyle \sigma_2(\cdot)=\frac{1}{2}(\rm tr \; \cdot)^2-\frac{1}{2}|\cdot|^2

the second elementary symmetric polynomial in its eigenvalues.

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