Related papers: Six dimensional counterexample to the Milnor Conje…

In this paper, we focus on Hamilton's pinching conjecture formulated in Hamilton's paper "Three-manifolds with positive Ricci curvature". Let $(M, g)$ be a complete, connected, noncompact Riemannian $3$-manifold satisfying the…

Let (M,d) be a metric space. For 0<r<R, and p in M let G(p,r,R) be the group obtained by considering all loops based at p whose image is contained in the closed ball of radius r and identifying two loops if there is a homotopy betweeen them…

In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: ''Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the…

We provide a step towards classifying Riemannian four-manifolds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs…

Let $(M,g)$ be a complete, connected, non-compact Riemannian three-manifold with non-negative Ricci curvature satisfying $Ric\geq\varepsilon\,\operatorname{tr}(Ric)\,g$ for some $\varepsilon>0$. In this note, we give a new proof based on…

We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially…

We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic…

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds…

In this paper, we show that for a sequence of orientable complete uniformly asymptotically flat $3$-manifolds $(M_i , g_i)$ with nonnegative scalar curvature and ADM mass $m(g_i)$ tending to zero, by subtracting some open subsets $Z_i$,…

Let $M$ be an open $n$-manifold with nonnegative Ricci curvature. We prove that if its escape rate is not $1/2$ and its Riemannian universal cover is conic at infinity, that is, every asymptotic cone $(Y,y)$ of the universal cover is a…

Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and…

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n$, for $3 \leq n \leq 7$, and non-negative Ricci curvature. Let $g = \phi^2 g_0$ be a metric in the conformal class of $g_0$. We show that there exists a smooth closed embedded…

We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector…

We find new conditions that the existence of nullity of the curvature tensor of an irreducible homogeneous space $M=G/H$ imposes on the Lie algebra $\mathfrak g$ of $G$ and on the Lie algebra $\tilde{\mathfrak g}$ of the full isometry group…

A smooth closed manifold $M$ is called almost Ricci-flat if $$\inf_g||\textrm{Ric}_g||_\infty\cdot \textrm{diam}_g(M)^2=0$$ where $\textrm{Ric}_g$ and $\textrm{diam}_g$ denote the Ricci tensor and the diameter of $g$ respectively and $g$…

We study the rigidity problems for open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature. We prove that if an asymptotic cone of $M$ properly contains a Euclidean $\mathbb{R}^{k-1}$, then the first Betti number of…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

We formulate a conjecture relating the topology of a manifold's universal cover with the existence of metrics with positive $m$-intermediate curvature. We prove the result for manifolds of dimension $n\in\{3,4,5\}$ and for most choices of…

We show that the space of metrics of positive scalar curvature on any 3-manifold is either empty or contractible. Second, we show that the diffeomorphism group of every 3-dimensional spherical space form deformation retracts to its isometry…

We survey the implications of our joint work with E. Bru\`e and A. Pigati on the structure of blow-downs for a smooth, complete, Riemannian $4$-manifold with nonnegative Ricci curvature and Euclidean volume growth. Very imprecisely, any…