# Ngô Quốc Anh

## April 19, 2011

### Flows for Q-curvature problems

Filed under: Riemannian geometry — Ngô Quốc Anh @ 4:14

At the time when I studied the flow method, I was confused whenever we use the evolution for metric or conformal factor. This is why I am writing this top to address this question.

For the convenience, we shall consider the flow for $Q$-curvature over an $n$-dimensional sphere $\mathbb S^n$. Let $g$ be a metric on $\mathbb S^n$ which is conformally equivalent to the standard metric $g_0$. If we write

$g=e^{nw}g_0$

then the Paneitz operator with respect to $g$ becomes

$P_g = e^{-nw}P_{g_0}$

and the $Q$-curvature of $g$ is given by

$Q_g=e^{-nw}(Q_{g_0}+P_{g_0}w).$

As the evolution of metric, the $Q$-curvature flow in defined as

$\displaystyle \frac{\partial}{\partial t}g(t)=-(Q_{g(t)}-\overline Q_{g(t)})g(t)$

where $\overline Q$ denotes the average of $Q$, that is

$\displaystyle \overline Q_{g(t)}=\frac{1}{\int_{\mathbb S^n} dv_{g(t)}}\int_{\mathbb S^n} Q_{g(t)}dv_{g(t)}.$

Suppose $g(t)$ is conformal to $g_0$, that is

$g(t)=e^{2w(t)}g_0$

we have

$\displaystyle\frac{\partial }{{\partial t}}g(t) = \frac{\partial }{{\partial t}}\left( {{e^{2w(t)}}{g_0}} \right) = 2{e^{2w(t)}}{g_0}\frac{\partial }{{\partial t}}w(t).$

Therefore,

$\displaystyle 2{e^{2w(t)}}{g_0}\frac{\partial }{{\partial t}}w(t) = \frac{\partial }{{\partial t}}g(t) = - ({Q_{g(t)}} - {{\overline Q}_{g(t)}})g(t) = - ({Q_{g(t)}} - {{\overline Q}_{g(t)}}){e^{2w(t)}}{g_0}$

which is equivalent to

$\displaystyle \frac{\partial }{{\partial t}}w(t) = - \frac{1}{2}({Q_{g(t)}} - {{\overline Q}_{g(t)}}).$

This is the $Q$-curvature flow written as the evolution of conformal factor $w$.