Ricci Flow on a Class of Noncompact Warped Product Manifolds

We consider the Ricci flow on noncompact n + 1 -dimensional manifolds M with symmetries, corresponding to warped product manifolds R × T n with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate | Rm | ( p , t ) ≤ C / t for some C > 0 and all p ∈ M , t ∈ ( 1 , ∞ ) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as t → ∞ ) while the curvatures stay uniformly bounded, and furthermore, the solution converges to a lower dimensional manifold. Moreover, if the ( n -dimensional) volumes of hypersurfaces coming from the symmetries of M are uniformly bounded, the solution converges locally uniformly to a flat cylinder after appropriate rescaling and pullback by a family of diffeomorphisms. Corresponding results are also shown for the normalized (i.e. volume preserving) Ricci flow.