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.

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