Convergence of Ricci Flow on a Class of Warped Product Metrics

We consider Ricci flow starting from warped product manifolds R × N , k 0 + g 0 2 g N , whose typical fibre ( N , g N ) is closed and Ricci flat. Here k 0 is a Riemannian metric on R and g 0 : R → R is positive. Under a mild condition, we show that (i) if the initial metric is asymptotic to the Ricci flat metric k 0 + c 2 g N , where c > 0 , the solution of the Ricci flow converges smoothly uniformly to a Ricci flat metric as t → ∞ , up to pullback by a family of diffeomorphisms, and (ii) if the initial manifold is asymptotic to the real line, then the solution converges uniformly (in Gromov Hausdorff distance) to the real line as t → ∞ . In the course of the proof, we establish an averaging and a convergence result for the heat equation on noncompact manifolds with time-dependent metric, that might be of independent interest.