The Ricci flow under almost non-negative curvature conditions

TL;DR

This paper extends Ricci flow invariant curvature conditions to cases with slightly negative bounds, showing that under certain conditions, the curvature remains controlled over short times, leading to new classification and smoothing results.

Contribution

It introduces generalized Ricci flow invariant conditions allowing for negative curvature bounds, with applications to manifold classification and smoothing of singular spaces.

Findings

Metrics with curvature operator eigenvalues > -1 can be evolved with controlled eigenvalues.

Generalizations apply to Kähler curvature conditions.

Results include classification of non-collapsed manifolds with near non-negative curvature.

Abstract

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 $-1$ 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 non-collapsedness. 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 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).