The Ricci flow under almost nonnegative curvature conditions
Abstract
We generalize most of the known Ricci flow invariant nonnegative 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 $1$ can be evolved by the Ricci flow for some uniform time such that the eigenvalues of the curvature operator remain greater than $C$. Here the time of existence and the constant $C$ only depend on the dimension and the degree of noncollapsedness. We obtain similar generalizations for other invariant curvature conditions, including positive biholomorphic curvature in the Kaehler 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 noncollapsed manifolds with almost nonnegative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds. We also obtain a shorttime existence result for the Ricci flow on open manifolds with almost nonnegative curvature (without requiring upper curvature bounds).
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.03002
 Bibcode:
 2017arXiv170703002B
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs
 EPrint:
 References added, changes in Cor 4 and in Remarks after