The Ricci flow under almost non-negative curvature conditions

• 成果类型: Article
• 署名作者: Bamler, Richard H.; Cabezas-Rivas, Esther; Wilking, Burkhard
• 署名单位: University of California System; University of California Berkeley; Goethe University Frankfurt; University of Munster
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00864-7
• 发表日期: 2019
• 页码: 95-126

摘要

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 Kahler 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).