Convergence of Ricci Flow on a Class of Warped Product Metrics

Abstract

We consider Ricci flow starting from warped product manifolds \(\left( {\mathbb {R}}\times N, k_0 + g_0^2 g_N\right) \), whose typical fibre \((N,g_N)\) is closed and Ricci flat. Here \(k_0\) is a Riemannian metric on \({\mathbb {R}}\) and \(g_0: {\mathbb {R}}\rightarrow {\mathbb {R}}\) 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 \(t \rightarrow \infty \), 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 \(t \rightarrow \infty \). 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.