# 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

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

## 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.