Convergence of Ricci Flow on a Class of Warped Product Metrics

Convergence of Ricci flow solutions to Taub-NUT

“…Since by Lemmas 2.10 and 4.2 c is uniformly bounded and spatially increasing, we see that c s ðÁ, tÞ is integrable for all t ! 0: Moreover, from (12) we derive that jc ss j aðtÞcðs, tÞ aðtÞm À1 g 0 in the space-time being the flow smooth for all positive times. Therefore c s ðs, tÞ !…”

“…Since the Ricci flow preserves isometries, one might consider looking for solutions converging to Ricci-flat fixed points when symmetries are present. In this direction, Marxen recently generalised earlier work of Hamilton to prove that if ðN, g N Þ is closed and Ricci-flat, then a class of warped product solutions to the Ricci flow ðR Â N, gðtÞÞ, of the form gðtÞ ¼ k 2 ðx, tÞdx 2 þ f 2 ðx, tÞg N , converge to ðR Â N, dx 2 þ c 2 g N Þ, for some c > 0, whenever the initial condition is asymptotic to the target Ricci-flat metric [12]. On the other hand, in the maximally symmetric case of homogeneous Ricci flows, convergence to Ricci-flat non-flat spaces is not possible due to a classic result of Alekseevskii and Kimelfeld [13].…”