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We will show the Cheeger-Colding segment inequality for manifolds with integral Ricci curvature bound. By using this segment inequality, the almost rigidity structure results for integral Ricci curvature will be derived by a similar method as in cite{CC1}. And the sharp Holder continuity result of cite{CoN} holds in the limit space of manifolds with integral Ricci curvature bound.
Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact K{a}hler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {em almost nonpositive $k$-Ricci curvature}, which is weaker than
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Chengs maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition.
We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature w
We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{
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