Optimal transport and Ricci curvature for metric-measure spaces

On the projective Ricci curvature

The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics.

Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons

In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.

Riemannian Submersions Need Not Preserve Positive Ricci Curvature

If $π:M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of (arbitrarily) negative Ricci curvature, but that there are no Riemannian submersions from manifolds with positive Ricci curvature to manifolds with nonpositive Ricci curvature.

Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature

We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that it is a quotient of $Σ\times \mathbb{C}^{n-1-k}\times N^k$, where $Σ$ and $N$ denote the Hamilton's cigar soliton and some compact Kähler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kähler Ricci soliton with $Ric\geq 0$, together with subquadratic volume growth or $\limsup_{r\to \infty} rS<1$ must have u…

Ricci-mean curvature flows in gradient shrinking Ricci solitons

Huisken studied asymptotic behavior of a mean curvature flow in a Euclidean space when it develops a singularity of type I, and proved that its rescaled flow converges to a self-shrinker in the Euclidean space. In this paper, we generalize this result for a Ricci-mean curvature flow moving along a Ricci flow constructed from a gradient shrinking Ricci soliton.

Shi-type estimates of the Ricci flow based on Ricci curvature

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\leq Ct^{-1}$ for all $t\in (0,T]$, where $C$ depends only on the dimension $n$ and $T$ depends on $K$ and the injectivity radius $inj_{g(t)}$. In the second part of this paper, we discuss the behavior of Ricci curvature and its derivative when the injectivity radius is thoroughly…

Weighted Projective Ricci Curvature in Finsler Geometry

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

In this paper, we study the injectivity radius bound for 3-d Ricci flow. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 of the book of Chow-Lu-Ni (2006).

Weak Ricci curvature bounds for Ricci shrinkers

We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.

On gradient Ricci solitons with constant scalar curvature

We use the theory of isoparametric functions to investigate gradient Ricci solitons with constant scalar curvature. We show rigidity of gradient Ricci solitons with constant scalar curvature under some conditions on the Ricci tensor, which are all satisfied if the manifold is curvature homogeneous. This leads to a complete description of four- and six-dimensional Kaehler gradient Ricci solitons with constant scalar curvature.

Classification of gradient steady Ricci solitons with linear curvature decay

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $κ$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

On Ricci curvature and volume growth in dimension three

We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

Steady Ricci solitons with horizontally $ε$-pinched Ricci curvature

In this paper, we prove that any $κ$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $ε$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $κ$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies $\lim_{r(x)\rightarrow\infty}R(x)f(x)=C_0\sup_{x\in M}R(x)$ with $C_0>\frac{n-2}{2}$.

A note on two-positive Ricci curvature and positive Einstein curvature

In this short note we discuss some recent results about two-positive Ricci curvature and their applications to positive Einstein curvature.

Metrics of positive Ricci curvature on bundles

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 bundle over a compact manifold of nonnegative Ricci curvature, then the product of E and R^p admits a complete metric of positive Ricci curvature for all large p.

Manifolds with A Lower Ricci Curvature Bound

This paper is a survey on the structure of manifolds with a lower Ricci curvature bound.

Curvature tensor under the Ricci-Harmonic flow

We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.

Uniqueness of asymptotic cones of complete noncompact shrinking gradient Ricci solitons with Ricci curvature decay

We discuss an elementary consequence of the works of (1) Brett Kotschwar and Lu Wang and (2) Ovidiu Munteanu and Jiaping Wang.

Weak stability of Ricci expanders with positive curvature operator

We prove the weak stability of expanding gradient Ricci solitons with positive curvature operator and quadratic curvature decay at infinity.

非负曲率的古老里奇流

Ancient Ricci flows with nonnegative Ricci curvature

本文研究了非负曲率非塌陷古老里奇流的渐近几何，通过其无穷远处的非塌陷F-极限度量孤子得出两个分歧定理。

On a dichotomy of the curvature decay of steady Ricci soliton

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset.

On the moment-angle manifolds with positive Ricci curvature

Riemannian metrics of positive Ricci curvature were constructed on certain moment-angle manifolds.

Manifolds with Pointwise Ricci Pinched Curvature

In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.

On the Ricci curvature of normal metrics on Biquotients

We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.

Convergence of Ricci-limit spaces under bounded Ricci curvature and local covering geometry I

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 $C^{1,α}$-regularities that are the best one may expect on those Ricci-limit spaces. As an application we prove an optimal generalization of Fukaya's fibration theorem on collapsed manifolds with bounded Ricci curvature, which also improves the original version to $C^{1,α}$ limit spaces.

On shrinking Gradient Ricci Soliton With Nonnegative Sectional Curvature

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.

Metrics of positive Ricci curvature on vector bundles over nilmanifolds

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 homeomorphic to manifolds of nonnegative sectional curvature.

On almost nonpositive $k$-Ricci curvature

Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {\em almost nonpositive $k$-Ricci curvature}, which is weaker than the existence of a Kähler metric with nonpositive $k$-Ricci curvature. When $k=1$, this is just the {\em almost nonpositive holomorphic sectional curvature} introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with $k$-Ricci curvature bounded from above by a pos…

Short-time persistence of bounded curvature under the Ricci flow

We use a first-order energy quantity to prove a strengthened statement of uniqueness for the Ricci flow. One consequence of this statement is that if a complete solution on a noncompact manifold has uniformly bounded Ricci curvature, then its sectional curvature will remain bounded for a short time if it is bounded initially. In other words, the Weyl curvature tensor of a complete solution to the Ricci flow cannot become unbounded instantaneously if the Ricci curvature remains bounded.

Ricci curvature for submanifolds of Bochner Kahler manifold

B. Y. Chen establish the relationship between the Ricci curvature and the squared mean curvature for submanifolds of Riemannian space form with arbitrary codimension. In this paper, we generalize the relationship between the Ricci curvature and the squared norm of mean curvature vector for submanifolds of Bochner Kahler manifolds.

A construction of graphs with positive Ricci curvature

Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positive Ricci curvature.

Improved Chen-Ricci inequality for curvature-like tensors and its applications

We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms and C-totally real submanifolds of Sasakian space forms.

Curvature Estimates for the Ricci Flow I

In this paper we present several curvature estimates for solutions of the Ricci flow which depend on smallness of certain local integrals of the norm of the Riemann curvature tensor.

Scalar curvature behavior of homogeneous Ricci flows

We prove that the scalar curvature of a homogeneous Ricci flow solution blows up at a forward or backward finite-time singularity.

Segment inequality and almost rigidity structures for integral Ricci curvature

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 Hölder continuity result of \cite{CoN} holds in the limit space of manifolds with integral Ricci curvature bound.

Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new feature is that no curvature upper bound is assumed.

Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infnite Betti numbers

In a previous paper, we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimension $\ge 6$. The purpose of the present paper is to use a different way to exhibit a family of complete $I$-dimensinal ($I\ge5$) Riemannian manifolds of positive Ricci curvature, quadratically asymptotically nonnegative sectional curvature, and certain infinite Betti number $b_j$ ($2\le j\le I-2$).

Mild Ricci curvature restrictions for steady gradient Ricci solitons

We show for a complete noncompact steady Ricci soliton that there exists a sequence {x_i} of points tending to infinity such that |Rc|(x_i) limits to zero.

纤维丛上的正中间里奇曲率

Positive Intermediate Ricci Curvature on Fibre Bundles

本文证明了浸没度量的典型变分型结果，结合手术技术为高连通流形和 exotic 球面等例子建立了正中间里奇曲率度量的存在性，并将正里奇曲率模空间结果推广至正中间情形。

具有2-Ricci曲率下界的3流形的体积比较与Ricci流

Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

本文研究2-Ricci曲率下界3流形的体积比较，建立测地球面面积和球体积的单调性与刚性，并推导预紧性、Ricci流下曲率下界的保持及非负标量曲率的拓扑刚性。

Curvature bounds via Ricci smoothing

We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.

A Heat Kernel Lower Bound for Integral Ricci Curvature

In this note we give a heat kernel lower bound in term of integral Ricci curvature, extending Cheeger-Yau's estimate.

Inner-Outer Curvatures, Ricci-Ollivier Curvature and Volume Growth of Graphs

We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.

Pseudolocality of Ricci Flow under Integral Bound of Curvature

We prove a pseudolocality type theorem for compact Ricci Flow under local integral bounds of curvature. The main tool is Local Ricci Flow introduced by Deane Yang in [4] and Pseudolocality Theorem of Perelman in [3]. We also study L^p bounds for the derivatives of curvature and smooth extension of Local Ricci Flow.

统计浸没的里奇曲率界

Ricci curvature bounds for Statistical Submersions

本文从直接里奇曲率视角，结合统计浸没的里奇曲率关系与Hineva代数不等式，首次得到统计浸没的希内瓦型下里奇曲率估计，为统计浸没提供统一里奇曲率方法并获得双侧里奇曲率界。

Small curvature concentration and Ricci flow smoothing

We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at positive times. As applications, we obtain under similar assumptions a compactness result and a gap theorem for complete noncompact manifolds with nonnegative Ricci Curvature.

Collapsing geometry with Ricci curvature bounded below and Ricci flow smoothing

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.

Sobolev inequalities on manifolds with nonnegative Bakry-Émery Ricci curvature

In this note we extend a recent result of S. Brendle [3] to Riemannian manifolds with densities and nonnegative Bakry-Émery Ricci curvature.

Comparison geometry of manifolds with boundary under a lower weighted Ricci curvature bound

We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

关于四维梯度Ricci孤子的曲率估计

On curvature estimates for four-dimensional gradient Ricci solitons

本文综述并比较四维梯度Ricci孤子的曲率估计，特别针对收缩子与扩张子，并为稳态孤子提供新的锐利估计$|Rm|\le C R$。

Compactness of Constant Mean Curvature Surfaces in Three Manifold with Positive Ricci Curvature

In this paper we prove a compactness theorem for constant mean curvature surfaces with area and genus bound in three manifold with positive Ricci curvature. As an application, we give a lower bound of first eigenvalue of constant mean curvature surfaces in three manifold with positive Ricci curvature.

Geometric inequalities for manifolds with Ricci curvature in the Kato class

We obtain an Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, Betti number estimates for closed manifolds whose Ricci curvature is controlled in the Kato class.

Boundary Effect of Ricci Curvature

On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then we discuss related rigidity questions and prove Ricci curvature rigidity results for manifolds with boundary.

On compact Riemannian manifolds with convex boundary and Ricci curvature bounded from below

We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results that support these conjectures are established.