# Ngô Quốc Anh

## 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.

Note that, under a conformal change of the metric

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

the quantity $Q$ transforms according to

$\displaystyle Q_{\widetilde g}=e^{-4u}(P_gu+Q_g)$

where $P=P_g$ denotes the Paneitz operator with respect to $g$.Keep in mind that the Paneitz operator is conformally invariant in the sense that

$\displaystyle P_{\widetilde g}=e^{-4u}P_g$

for any conformal metric

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

For any $g$, the operator $P_g$ acts on a smooth function $u$ on $M$ via

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

which plays a similar role as the Laplace operator in dimension two.

We know from the Gauss-Bonnet-Chern theorem that

$\displaystyle \int_M {Qd{v_g}} + \frac{1}{4}\int_M {|W{|^2}d{v_g}} = 8{\pi ^2}\chi (M)$

where $W$ denotes the Weyl tensor. It now follows from the fact that $W$ is conformally invariant that

$\displaystyle k = {k_g} = \int_M {Qd{v_g}}$

is also conformally invariant. Indeed, this can be seen from the rule developed in this note.

The prescribed $Q$-curvature problem can be formulated as follows:

Whether on a given four-manifold $(M,g)$ there exists a conformal metric $\widetilde g =e^{2u}g$ for which the $Q$-curvature of $\widetilde g$ equals the prescribed function $\widetilde Q$?

In terms of PDE, this is related to solving the following equation

$\displaystyle P_gu+2Q_g=2\widetilde Q e^{4u}$.

This equation is the Euler-Lagrange equation of the functional

$\displaystyle \mathcal F = \int_M {u{P_g}ud{v_g}} + 4\int_M {{Q_g}ud{v_g}} - \left( {\int_M {{Q_g}d{v_g}} } \right)\log \left( {\int_M {\widetilde Q{e^{4u}}d{v_g}} } \right).$

A lot of progresses have been done on this problem.