Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

Singular Ricci Solitons and Their Stability under the Ricci Flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow “pushes away” from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. The main objective of this paper is to study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric. We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies that the “opening up” of the singularity persists for the perturbations as well.     

Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four

We obtain a local smoothing result for Riemannian manifolds with bounded Ricci curvatures in dimension four. More precisely, given a Riemannian metric with bounded Ricci curvature and small L²-norm of curvature on a metric ball, we can find a smooth metric with bounded curvature which is C1,α-close to the original metric on a smaller ball but still of definite size.     

First eigenvalue monotonicity for the p-Laplace operator under the Ricci flow

In this note, we discuss the monotonicity of the first eigenvalue of the p-Laplace operator (p ?? 2) along the Ricci flow on closed Riemannian manifolds. We prove that the first eigenvalue of the p-Laplace operator is nondecreasing along the Ricci flow under some different curvature assumptions, and therefore extend some parts of Ma??s results [Ann. Glob. Anal. Geom., 29, 287?C292 (2006)].     

Curvature integrals under the Ricci flow on surfaces

In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.      

Bakry–Emery Curvature Operator and Ricci Flow

In this paper, we introduce some techniques of Bakry–Emery curvature operator to Ricci flow and prove the evolution equation for the Bakry–Emery scalar curvature. As its application, we can easily derive the Perelman’s entropy functional monotonicity formula. We also discuss some gradient estimates of Ricci curvature and L ²– estimates of scalar curvature.Project partially supported by Yumiao Fund of Putian University.     

Classification of gradient steady Ricci solitons with linear curvature decay

In this paper,we study steady Ricci solitons with a linear decay of sectional curvature.In particular,we give a complete classification of 3-dimensional steady Ricci solitons and 4-dimensional K-noncollapsed steady Ricci solitons with non-negative sectional curvature under the linear curvature decay.     

L 2 Forms and Ricci Flow with Bounded Curvature on Complete Non-Compact Manifolds

In this paper, we study the evolution of L ² one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the L ² norm of a smooth one form is non-increasing along the Ricci flow with bounded curvature. The L ^∞ norm is showed to have monotonicity property too. Then we use L ^∞ cohomology of one forms with compact support to study the singularity model for the Ricci flow on .     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

Ricci solitons on compact Kähler surfaces

We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming that the curvature is slightly more positive than that of the single known example of a soliton in this dimension.

     

Normalized Ricci flow on nonparabolic surfaces

This paper studies the normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically –1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature –1. A relative estimate of Green’s function is proved as a tool.      

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

Remark on a lower diameter bound for compact shrinking Ricci solitons

In this paper, inspired by Fernández-López and García-Río [11], we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13], we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.     

Ricci curvature of Markov chains on metric spaces

We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally extends to any Markov chain on a metric space. For a Riemannian manifold this gives back, after scaling, the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein-Uhlenbeck process. Moreover this generalization is consistent with the Bakry-Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold.Positive Ricci curvature is shown to imply a spectral gap, a Lévy-Gromov-like Gaussian concentration theorem and a kind of modified logarithmic Sobolev inequality. The bounds obtained are sharp in a variety of examples.     

The second Sobolev best constant along the Ricci flow

In this work we present some properties satisfied by the second L ²-Riemannian Sobolev best constant along the Ricci flow on compact manifolds of dimensions n ≥ 4. We prove that, along the Ricci flow g(t), the second best constant B ₀(2, g(t)) depends continuously on t and blows-up in finite time. In certain cases, the speed of the explosion is, at least, the same one of the curvature operator. We also show that, on manifolds with positive curvature operator or pointwise 1/4-pinched curvature, one of the situations holds: B ₀(2, g(t)) converges to an explicit constant or extremal functions there exists for t large.      

Immortal solution of the Ricci flow

For any complete noncompact Kahler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow.     

Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature

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.      

关于非紧流形上的Ricci流的一个注记

设（M^3,90）是非紧三维Riemann流形,其Ricci曲率非负,单射半径有正的下界,且当x→∞时数量曲率R（x）→0。则以（M^3,go）为初始值的Ricci流在M^3×[0,∞）上有长期解。这推广了马和朱最近的一个结果．在高维情形我们也有相应的结果,并且我们给Chau,Tam和Yu在Ktihler情形的类似定理一个新的证明。