相关论文: Pinching estimates for solutions of the linearized…
We present local estimates for solutions to the Ricci flow, without the assumption that the solution has bounded curvature. These estimates lead to a generalisation of one of the pseudolocality results of G.Perelman in dimension two.
Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as…
In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary,…
This note illustrates the Ricci flow method based on the Cao.H.D's paper[1] and Yau.S.T's paper[4], and tries to explain the method in detail, especially in some calculations. Jian Song and Weinkove's note[9] used some other estimates to…
This is the first of a series of papers, where we introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions to the Ricci flow in the…
In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the…
In this note, we give a diffeomorphism (to $\mathbb{R}^n$) criterion via long-time Ricci flow and show some applications. In particular, we provide an affirmative answer that the conclusion in [Manifolds with small curvature concentration,…
We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the…
Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh…
We consider ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq \gamma_1 |H|^2, |\nabla^2 A| \leq \gamma_2 |H|^3$.…
In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under…
An important and natural question in the analysis of Ricci flow singularity formation in dimensions four and above is as follows: What are the weakest conditions that provide control of the norm of the Riemann curvature tensor? In this…
We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons.
We develop some estimates under the Ricci flow and use these estimates to study the blowup rates of curvatures at singularities. As applications, we obtain some gap theorems: $\displaystyle \sup_X |Ric|$ and $\displaystyle \sqrt{\sup_X…
We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to…
In this paper, we give the full proof of a conjecture of R.Hamilton that for $(M^3, g)$ being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\geq \ep R g$, where $R>0$ is the positive scalar…
For each $n\ge 3$, we construct a 'pancake-like', $O(2)\times O(n-1)$-invariant ancient Ricci flow with positive curvature operator and bounded "girth", and we determine its asymptotic limits backwards in time. This solution is new even in…
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…
For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality…
For all dimensions $n\geq5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that $\nabla^2f$ is $2-$nonnegative and the curvature tensor is WPIC1 at…