Related papers: Ricci Curvature, Diameter and Fundamental groups

We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called "curvature concentration"), by showing that the…

We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not…

In this paper we study the topology of compact manifolds of positive isotropic curvature (PIC). There are many examples of non-simply connected compact manifolds with positive isotropic curvature. We prove that the fundamental group of a…

A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature…

We investigate the curvature of invariant metrics on G-manifolds with finitely many non-principal orbits. We prove existence results for metrics of positive Ricci curvature and non-negative sectional curvature, and discuss some families of…

We present conditions on the Ricci curvature for complete, oriented, minimal submanifolds of Euclidean space, as well as the standard unit sphere, when the Gauss maps are bounded embeddings.

In my talk I will discuss the following results which were obtained in joint work with Wilderich Tuschmann. 1. For any given numbers $m$, $C$ and $D$, the class of $m$-dimensional simply connected closed smooth manifolds with finite second…

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci…

We first extend Cheeger-Colding Almost Splitting Theorem to smooth metric measure spaces. Arguments utilizing this extension of the Almost Splitting Theorem show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci…

The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs.…

It is proved that the fundamental group of a complete Riemannian manifold with nonnegative Ricci curvature and certain volume growth conditions is trivial or finite.

We extend Cheeger-Gromov's and Anderson's convergence theorems to regular limit spaces of manifolds with bounded Ricci curvature and local covering geometry, by establishing the $C^{1,\alpha}$-regularities that are the best one may expect…

This paper is devoted to a deep analysis of the process known as Cheeger deformation, applied to manifolds with isometric group actions. Here, we provide new curvature estimates near singular orbits and present several applications. As the…

The paper is devoted to Hardy type inequalities on closed manifolds. By means of various weighted Ricci curvatures, we establish several sharp Hardy type inequalities on closed weighted Riemannian manifolds. Our results complement in…

We construct metrics of positive $2^{\rm nd}$ intermediate Ricci curvature, $\mathrm{Ric}_2>0$, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including $\mathbb{S}^6\times\mathbb{S}^7$, $\mathbb{S}^7\times\mathbb{S}^7$ and all…

The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds…

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature with additional stability condition on $\widetilde{M}$, the Riemannian universal cover of $M$. We prove that if any tangent cone of $\widetilde{M}$ at…

We study integral and pointwise bounds on the curvature of gradient shrinking Ricci solitons. As applications we discuss gap and compactness results for gradient shrinkers.

We give examples of pinched negatively curved manifolds for which the Ricci flow does not converge smoothly.

We study orientability in spaces with Ricci curvature bounded below. Building on the theory developed by Honda, we establish equivalent characterizations of orientability for Ricci limit and RCD spaces in terms of the orientability of their…