Ricci flow of locally homogeneous geometries on closed manifolds (Q1203631)

A Riemannian metric \(g\) on a three-manifold \(M^ 3\) is defined to be locally homogeneous if, for every pair of points \(x, y\in M^ 3\), there exist neighbourhoods \(U_ x\) of \(x\) and \(V_ y\) of \(y\) such that there is an isometry \(\Psi\) mapping \((U_ x, g_{/U_ x})\) \((V_ y, g_{/V_ y})\), with \(\Psi(x)=y\). \textit{W. P. Thurston} [Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982; Zbl 0496.57005)]claims that any closed three-manifold \(M^ 3\) may be canonically decomposed into pieces such that each of the pieces admits a locally homogeneous geometry. The idea for proving this conjecture is to use Ricci flow, i.e., to choose an arbitrary metric on \(M^ 3\) and then deform this metric via the Ricci flow equation \(\frac \partial {\partial t}g = -2\operatorname{Ric}+\frac 23\langle R\rangle g\), where \(\langle R\rangle\) denotes the average of the scalar curvature \(R\) over \(M^ 3\). In this paper, the authors provide a complete picture of the behavior of the Ricci flow of locally homogeneous Riemannian metrics on closed manifolds.