Evolution of an extended Ricci flow system

We show that Hamilton’s Ricci flow and the static Einsteinvacuum equations are closely connected by the followingsystem of geometric evolution equations:\begin{align*}\tdel g &={-}2Rc(g) + 2\alpha_ndu\otimes du,\tdel u &= \Delta^{g}u,\end{align*}where $g(t)$ is a Riemannian metric, $u(t)$ a scalar function and $\alpha_n$ aconstant depending only on the dimension $n\geq 3$. This provides aninteresting and useful link from problems in low-dimensional topology andgeometry to physical questions in general relativity.

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Published 1 January 2008