On the long time behavior of homogeneous Ricci flows (Q894374)

A Ricci flow solution is called a homogeneous Ricci flow solution if the initial metric \(g_0\) is homogeneous. In this case all evolved metrics are homogeneous too and the isometry groups do not change. If a solution to the Ricci flow cannot be extended smoothly after time \(T\), then \(T \in (0, \infty]\) is called a singular time. If the singular time \(T\) is finite, the Ricci flow solution is said to have a finite extinction time. For a Ricci flow solution there are notions of special types of singularities (types I--III). In this paper some structure results for homogeneous Ricci flow solutions are proved. Among them are the following: {\parindent=0.5cm \begin{itemize}\item[--] Any homogeneous Ricci flow solution with finite extinction time develops a type-I singularity. \item[--] Any homogeneous Ricci flow solution on a compact homogeneous space, not diffeomorphic to a torus, has finite extinction time. \item[--] Any immortal homogeneous Ricci flow solution develops a type-III singularity. \end{itemize}} Let \(M\) be some homogeneous spaces with compact or Euclidean universal covering. Then any homogeneous Ricci flow on \(M\) (with an arbitrary initial metric) develops either a type-I or a type-III singularity. Also some old results of S. Bochner and A. Spiro about Ricci curvature of locally homogeneous spaces are reproved.