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In this paper we prove a conjecture by Feldman-Ilmanen-Knopf in \cite{FIK} that the gradient shrinking soliton metric they constructed on the tautological line bundle over $\CP^1$ is the uniform limit of blow-ups of a type I Ricci flow…

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…

In this note, we prove that for a complete noncompact three dimensional Riemannian manifold with bounded sectional curvature, if it has uniformly positive scalar curvature, then there is a uniform lower bound on the injectivity radius.

The present paper deals with the proper existence of a generalized class of recurrent manifolds, namely, hyper-generalized recurrent manifolds. We have established the proper existence of various generalized notions of recurrent manifolds.…

We study the fundamental group of an open $n$-manifold $M$ of nonnegative Ricci curvature. We show that if there is an integer $k$ such that any tangent cone at infinity of the Riemannian universal cover of $M$ is a metric cone, whose…

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive…

In this paper we generalize the theory of Cheeger, Colding and Naber to certain singular spaces that arise as limits of sequences of Riemannian manifolds. This theory will have applications in the analysis of Ricci flows of bounded…

We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci…

In this paper we study the pseudolocality theorems of Ricci flows on incomplete manifolds. We prove that if a ball with its closure contained in an incomplete manifold has the small scalar curvature lower bound and almost Euclidean…

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time…

We construct bi-Lipschitz embeddings into Euclidean space for manifolds and orbifolds of bounded diameter and curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. Our results also…

Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. Associated with $L$ one has \textit{le…

In this note we will show the almost maximal volume entropy rigidity for manifolds with lower integral Ricci curvature bound in the non-collapsing case: Given $n, d, p>\frac{n}{2}$, there exist $\delta(n, d, p), \epsilon(n, d, p)>0$, such…

The energy of any $C^1$ representative of a homotopy class of maps from a compact and connected Riemannian manifold with nonnegative Ricci curvature into a complete Riemannian manifold with no conjugate points is bounded below by a constant…

We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.

In this paper, we present extensions of the classical Bonnet-Myers theorem for Riemannian manifolds with nonnegative Ricci curvature. Our results provide criteria for compactness and a method for estimating the diameter of such manifolds…

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for…

The Colding-Gromov gap theorem asserts that an almost non-negatively Ricci curved manifold with unit diameter and maximal first Betti number is homeomorphic to the flat torus. In this paper, we prove a parametrized version of this theorem,…

Let $M$ denote a low-dimensional manifold embedded in Euclidean space and let ${X}= \{ x_1, \dots, x_n \}$ be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph…

We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\geq 2$ has $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq 0$ for some…