Complete ancient solutions to the Ricci flow with pinched curvature

We show that any complete ancient solution to the Ricci flow equation with possibly unbounded curvature has constant curvature at each time if its curvature is pinched all the time. This is a slight extension of a result of Brendle, Huisken and Sinestrari for ancient solutions on compact manifolds. In our proof, we adapt their argument relying on the maximum principle with the help of Chen’s technique.

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Received 1 September 2014

Published 4 August 2017