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.

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