The Ricci flow under almost non-negative curvature conditions

By Richard Bamler, Esther Cabezas-Rivas, Burkhard Wilking. Inventiones Mathematicae, 2017

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We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $$-\,1$$-1 can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $$-\,C$$-C. Here the time of existence and the constant C only depend on the dimension and the degree of non-collapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kähler case. We also get a local version of the main theorem. As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a short-time existence result for the Ricci flow on open manifolds with almost non-negative curvature (without requiring upper curvature bounds).

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

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