We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly PIC 1 cases, of which the results are new. We complete the discussion of the almost preservation problem by Bamler-Cabezas-Rivas-Wilking, and the 2-non-negative case generalizes a result in 3D by Simon-Topping to higher dimensions.As an application, we use the local Ricci flow to smooth a metric space which is the limit of a sequence of manifolds with the almost non-negative curvature conditions, and show that this limit space is bi-Hölder homeomorphic to a smooth manifold.
Hamilton conjectured that there exists a 3d steady gradient Ricci soliton that is a flying wing. We confirm his conjecture in this paper. We also found a family of Z 2 × O(n − 1)-symmetric n-dimensional steady gradient Ricci solitons with positive curvature operator in all dimension n ≥ 3.
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