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 (mathbb {R}times T^n) with flat fibres. We show longtime existence and that the Ricci flow solution is of type III, i.e. the curvature estimate (|{{mathrm{Rm}}}|(p,t) le C/t) for some (C > 0) and all (p in M, t in (1,infty )) holds. We also show that if M has finite volume, the solution collapses, i.e. the injectivity radius converges uniformly to 0 (as (t rightarrow infty )) 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.