Space of Ricci flows (I)
Xiuxiong Chen, Bing Wang
本文研究具有有界能量和数量曲率的非坍缩里奇流解的模空间。我们证明了这种模空间的一个弱紧定理，并应用它研究了等周常数控制、Kähler里奇流和梯度收缩孤子的模空间.
In this paper, we study the moduli spaces of noncollapsed Ricci flow solutions with bounded energy and scalar curvature. We show a weak compactness theorem for such moduli spaces and apply it to study isoperimetric constant control, Kähler Ricci flow and moduli space of gradient shrinking solitons.
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We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3d complete Riemannian manifold with non-negative Ricci curvature, there exists a smooth Ricci flow starting from it.
Ricci Lower Bound for Kähler-Ricci Flow
本文讨论了闭流形上具有奇异性的Kähler-里奇流的里奇下界.
Ricci-mean curvature flows in gradient shrinking Ricci solitons
Huisken研究了欧几里得空间中的中曲率流在发生I型奇点时的渐近行为，证明了其重标度流在欧几里得空间中收敛于自收缩流.本文将这一结果推广到沿由梯度收缩Ricci孤子构成的里奇流运动的Ricci-中曲率流。
In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton's Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.
Ricci Flow on 3-dimensional Lie groups and 4-dimensional Ricci-flat manifolds
研究了三维李群上的里奇流与四维里奇平坦流形的关系。特别地，我们构造了关于三维李群的Ricci平坦上齐性1度量。
Injectivity radius bound of Ricci flow with positive Ricci curvature and applications
本文研究了三维里奇流的内射半径界。作为应用，我们证明了具有正里奇曲率的里奇流的长时间存在性.我们还部分解决了Chow-Lu-Ni（2006）书中第302页的一个问题。
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。
Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures
本文研究Ricci流与Ricci-调和流下抛物频率的单调性，分别针对有界Bakry-Émery Ricci曲率和有界Ricci曲率两种情况，推广了相关结果。
On the conditions to extend Ricci flow(III)
本文对曲率估计进行了简化和改进：关于推广里奇流的条件（II）.此外，我们还对具有有界数量曲率的里奇流给出了一些体积估计。这些估计可用于研究里奇流的奇异性和Kähler里奇流的收敛性.
Stability of solutions of certain extended Ricci flow systems
考虑了四个推广的里奇流系统--即与其它几何流耦合的里奇流--并证明了这些流的某些定常解的动力稳定性。这些系统包括与调和映射流耦合的里奇流（在翘曲积上的里奇流的背景下抽象地研究）、与调和映射流和Yang-Mills流耦合的里奇流以及与度量相容连接的挠率的热流耦合的里奇流。用于证明稳定性的方法遵循Guenther、Isenberg和Knopf概述的程序，该程序使用拟线性抛物系统的最大正则性理论和Simonett的结果。
In this paper we have introduced the notion of $*-$ Ricci flow and shown that $*-$ Ricci soliton which was introduced by Kakimakamis and Panagiotid in 2014, is a self similar soliton of the $*-$ Ricci flow. We have also find the deformation of geometric curvature tensors under $*-$ Ricci flow. In the last two section of the paper, we have found the $\Im$-functional and $ω-$ functional for $*-$ Ricci flow respectively.
Positivity of Ricci curvature under the Kähler--Ricci flow
对所有复维数n>=2，我们构造了具有有界曲率和非负里奇曲率的完备Kaehler流形，其Kaehler-Ricci演化立即获得混合号的里奇曲率.
In this paper we compute the Ricci flow formulas for invariant metrics on prinicpal $G$-bundles compatible with the connection. Our primary focus is on torus bundles which we use to study a notion of Bakry-Émery Ricci flow as well as Ricci flow on circle bundles over Kähler-Einstein manifolds. The latter application gives us solutions to Ricci flows on Heisenberg groups and implicit solutions to Ricci flows on Berger 3-spheres and several other 3-dimensional manifolds.
Evolution of Ricci scalar under Finsler Ricci flow
最近，我们研究了一类Finsler度量沿Finsler里奇流的演化，并证明了它在短时间内的收敛性.本文给出了Finslerian里奇流的约化$hh$ -曲率和Ricci标量的发展方程，证明了里奇流在有限时刻保持约化$hh$ -曲率的正性.其次，证明了Ricci标量的演化是一个抛物型方程，并且如果初始Finsler度量具有正的旗曲率，则只要解存在，旗曲率和Ricci标量都保持为正.最后，得到了沿里奇流的里奇标量的下界。
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.
Existence, Lifespan and Transfer Rate of Ricci Flows on Manifolds with Small Ricci Curvature
我们发现，在4维及以上，里奇流的寿命取决于相对较小的里奇曲率相比，黎曼曲率的初始流形。我们可以将这个寿命估计推广到局部里奇流，利用它证明了曲率至多为二次增长的非紧黎曼流形上里奇流解的短时存在性，其中在黎曼曲率大的区域，其里奇曲率及其前两阶导数足够小.这些里奇流解可能具有无界曲率。此外，我们的方法意味着，在一些适当的假设下，空间传输率（高曲率区域影响低曲率区域的速率）的里奇流类似于热方程。
Ricci flow on modified Riemann extensions
研究了修正Riemann扩张在里奇流下的性质。得到了里奇流下的修正Riemann延拓仍然是修正Riemann延拓的充要条件。我们还讨论了曲率张量在里奇流下的性质。
A Modified Kähler-Ricci Flow
本文介绍并研究了一种修正的Kähler-里奇流。重点是展示Kähler-里奇流的灵活性，并总结一些有用的技巧。
We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. We consider the behavior of Ricci flow with surgery starting from a fixed initial compact Riemannian 3-manifold, as the surgery parameter varies. We prove that the flow with surgery subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.
A Note on Conformal Ricci Flow
本文研究了Arthur Fischer提出的共形里奇流。我们利用DeTurck的技巧将共形里奇流改写为一个强抛物-椭圆型偏微分方程。然后证明了紧流形和渐近平坦流形上共形里奇流的短时存在性。我们证明了Yamabe常数沿着紧致流形上的共形里奇流单调增加。我们还证明了共形里奇流是渐近平坦流形上ADM质量的梯度流。
The conical Kähler-Ricci flow on Fano manifolds
本文研究了Fano流形$M$上锥形Kähler-里奇流的长期行为.首先，在扭曲Kähler-Ricci流的局部一致正则性的基础上，通过限制一系列扭曲流，我们得到了锥形Kähler-里奇流的长时间解.其次，研究了扭曲Kähler-里奇流的一致Perelman估计.证明了锥Kähler-里奇流收敛于锥Kähler-Einstein度规的条件。
Smooth approximation of the modified conical Kähler-Ricci flow
我们引入了由全纯向量场修正的锥形Kähler-里奇流。我们构造了修正的锥形Kähler-里奇流的长时间解作为光滑Kähler-里奇流序列的极限.
In this paper, we prove the existence of a Kahler Ricci soliton on any smooth Fano horospherical manifold by a study of the Kahler-Ricci flow. Indeed, we prove that the renormalized Kahler Ricci flow converges in the sense of Cheeger Gromov and that this limit is a Kahler-Ricci soliton.
Eigenvalues and entropies under the harmonic-Ricci flow
本文讨论调和-里奇流，即与调和映射流耦合的里奇流下的特征值和熵。我们给出了另一种证明的结果，紧凑的稳定和扩大谐波里奇呼吸。在第二部分中，我们得到了调和-里奇流下Laplacian特征值的单调性公式。最后，我们得到了调和里奇流的收缩熵和扩张熵的一阶变分。
Short-time existence of the Ricci flow on complete, non-collapsed $3$-manifolds with Ricci curvature bounded from below
证明了对任意完备三流形，若其里奇曲率和半径为1的球的体积都有下界，则里奇流的解在短时间内存在.事实上，我们的证明也产生了一个（非规范）的方式来流动和正则化的一些内部区域的非完整的初始数据满足上述范围。
We investigate Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities and present evidence in favor of a conjecture that parabolic rescalings at the singularities converge to singularity models that are shrinking Kahler-Ricci solitons. Specifically, the singularity model for these solutions is expected to be the "blowdown soliton" discovered in [FIK03]. Our partial results support the conjecture that the blowdown soliton is stable under Ricci flow, as well as the conjectured stability of the subspace of Kahler metrics under Ricci flow.
In this paper, we consider Ricci flows admitting closed and smooth tangent flows in the sense of Bamler [Bam20c]. The tangent flow in question can be either a tangent flow at infinity for an ancient Ricci flow, or a tangent flow at a singular point for a Ricci flow developing a finite-time singularity. Among other things, we prove: (1) that in these cases the tangent flow must be unique, (2) that if a Ricci flow with finite-time singularity has a closed singularity model, then the singularity is of Type I and the singularity model is the tangent flow at the singular point; this answers a quest…
On the stability of harmonic maps under the homogeneous Ricci flow
本文研究了调和映射在齐次里奇流下的稳定性和不稳定性。我们提供的例子中的稳定性（非稳定性）下保持里奇流和一个例子中的里奇流不保持稳定的调和映射。
We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a $C^{0}$ orbifold at any finite-time singularity, so has an extension through the singularity via orbifold Ricci flow. Moreover, a Type-I blowup of the flow based at any orbifold point converges to a flat cone in the Gromov-Hausdorff sense, without passing to a subsequence. In addition, we prove $L^{p}$ bounds for the curvature tensor on time-slices for any $p<2$. In higher dimensions, we show that every singular point of …
Ricci Flow of Compact Locally Homogeneous Geometries on 5-Manifolds
本计画旨在分析五维流形中里奇流的行为。里奇流由理查德·汉密尔顿于1982年提出，是证明几何化和庞加莱猜想的重要工具。一般情况下，里奇流是一个非线性偏微分方程，其解是相当困难的计算;然而，在齐次流形，里奇流减少到常微分方程。里奇流在二维、三维和四维齐次流形中的行为已经被计算出来，并得到了很好的理解。本文的工作将描述如何更好地理解里奇流在某一类五维齐次流形中的行为。
On the weighted forward reduced Entropy of Ricci flow
本文首先介绍了里奇流的加权前向约化体积。与里奇流的扩张子有关的加权向前约化体积在非紧流形上得到了定义，并且在里奇流下单调非增.此外，我们证明了与Perelman约化体积相同，加权约化体积熵具有$(4π)^{\frac{n}{2}}$的充要条件是里奇流是平坦欧几里得空间上的平凡流.
In this article we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.
Pseudolocality of Ricci Flow under Integral Bound of Curvature
证明了紧致里奇流在曲率的局部积分界下的一个伪局部性定理。主要工具是Deane Yang在[4]中引入的局部里奇流和Perelman在[3]中的伪局部性定理。我们还研究了局部里奇流的曲率导数和光滑延拓的L^p界.
On the Conditions to Extend Ricci Flow
沿着闭流形上的里奇流解，我们证明了如果里奇曲率从下到上是一致有界的，那么一个数量曲率积分界就足以扩展流.而且，这种积分边界条件在某种意义上是最优的。
We show that for certain locally collapsing initial data with Ricci curvature bounded below, one could start the Ricci flow for a definite period of time. This provides a Ricci flow smoothing tool, with which we find topological conditions that detect the collapsing infranil fiber bundles over controlled Riemannian orbifolds among those locally collapsing regions with Ricci curvature bounded below. In the appendix, we also provide a local distance distortion estimate for certain Ricci flows with collapsing initial data.
An Elementary Remark on a Lower bound for the $μ$-Invariant of Singularity Models for Ricci Flow
里奇流奇异性模型的$μ$不变量的解释性观察。
Generalized Ricci flow II: existence for noncompact complete manifolds
本文继续研究广义里奇流。给出完备非紧黎曼流形上的稳定梯度Ricci孤子是Ricci平坦的一个判据，并引入一个自然流，其稳定点是Ricci平坦度量.修正了Shi和List的论证，证明了该模型的短时存在性和高阶导数估计。
We establish several quantitative results about singular Ricci flows, including estimates on the curvature and volume, and the set of singular times.
Remarks of weak-compactness along Kahler Ricci flow
给出了Fano流形上Kähler里奇流上时间片弱紧性的一个直接证明.
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…
On the dynamical behaviour of the generalized Ricci flow
本文将Müller-Haslhofer关于Ricci流下Ricci平坦度量动力学稳定性与不稳定性的结果推广到广义Ricci流，得到了Ricci平坦度量与消失3-形式配对的相应动力学稳定性与不稳定性结论。
Here, we study the existence and uniqueness of solutions to the Ricci flow on Finsler surfaces and show short time existence of solutions for such flows. To this purpose, we first study the Finslerian Ricci-DeTurck flow on Finsler surfaces and find a unique short time solution to this flow. Then, we find a solution to the original Ricci flow by pulling back the solution of the Ricci-DeTurck flow using appropriate diffeomorphisms. In the end, we illustrate this argument with some examples.
Curvature tensor under the Ricci-Harmonic flow
证明了若里奇曲率在Ricci-Harmonic流下对所有时间$t$ \in [0，T）都是一致有界的，则曲率张量也必须是一致有界的.
Hyperbolic thermostat and Hamilton's Harnack inequality for the Ricci flow
本文从双曲恒温器的角度恢复了里奇流的Hamilton Harnack不等式。
Scalar curvature behavior of homogeneous Ricci flows
证明了齐次里奇流解的数量曲率在向前或向后的有限时间奇点处爆破。
The logarithmic Sobolev inequality along the Ricci flow in dimension 2
本文给出了二维里奇流上对数Sobolev不等式的一些结果。
A short proof of the convergence of the Kahler-Ricci flow on Fano manifolds admitting a Kahler-Einstein metric or a Kahler-Ricci soliton is given, using a variety of recent techniques
Pseudoconnections and Ricci flow
在这篇笔记中，我们解释了如何流在空间的黎曼度量（包括里奇的\cite{mt}）诱导一个在空间的伪连接。
The Ricci Flow on Complete Noncompact Kähler Manifolds
本文综述了完备非紧Kähler流形上的里奇流及其几何应用的最新进展。
Consider a Riemannian manifold $(M^{m}, g)$ whose volume is the same as the standard sphere $(S^{m}, g_{round})$. If $p>\frac{m}{2}$ and $\int_{M} \left\{ Rc-(m-1)g\right\}_{-}^{p} dv$ is sufficiently small, we show that the normalized Ricci flow initiated from $(M^{m}, g)$ will exist immortally and converge to the standard sphere. The choice of $p$ is optimal.
In this work, we first establish short time existence and Shi's type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of Kaehler-Einstein metric on complete noncompact Hermitian manifolds.
The Sasaki-Ricci flow and compact Sasakian manifolds of positive transverse holomorphic bisectional curvature
我们证明了Perelman的W-泛函可以推广到Sasaki-里奇流。当基本第一Chern类为正时，我们证明了Sasaki-里奇流上横Ricci势的数量曲率、直径的一致界和$C^1$一致界，从而将Perelman关于Kahler-里奇流的结果推广到Sasaki情形.利用Bando和Mok在Kahler背景下证明的方法和结果，我们还证明了全纯双截曲率的横截的正性沿流保持不变.特别地，我们证明了Sasaki-里奇流将收敛到Sasaki-Ricci孤子，如果初始度量具有非负的横观全纯二分曲率。
Ricci Flow Equation on (α, β)-Metrics
本文研究满足非正规或正规里奇流方程的Finsler度量类，即（α，β）-度量.
Uniqueness of Conformal Ricci Flow using Energy Methods
我们分析了沿具有常负数量曲率的闭流形的共形里奇流的能量泛函。给定初始条件，我们使用这个功能来证明的唯一性的度量和压力函数沿共形里奇流。
The twisted Kahler-Ricci flow
本文研究Kahler-里奇流的一个推广，其中Ricci形式被一个封闭的非负（1，1）-形式扭曲.我们表明，当扭曲的Kahler-Einstein度量存在，那么这个扭曲流指数收敛。这推广了Perelman关于Kahler-里奇流收敛性的一个结果，并且建立在Tian-Zhu的工作基础上。
The Chern-Ricci flow on smooth minimal models of general type
我们证明了在一般型光滑Hermitian极小模型上，Chern-里奇流收敛于M上的一个闭正流。此外，流光滑收敛到Kahler-Einstein度量的紧集远离零轨迹的K_M。这将Tsuji和Tian-Zhang的工作推广到Hermitian流形，进一步证明了陈-里奇流是Kahler-里奇流的自然推广。
The Long-Time Behavior of the Ricci Tensor Under The Ricci Flow
我们证明了，给定曲率和直径一致有界的闭流形上的里奇流的不朽解，当t趋于无穷大时，里奇张量趋于零。我们还证明了，如果在一个闭的三维流形上存在一个不朽解，使得直径的平方与曲率张量的范数的乘积是一致有界的，那么这个解一定是III型的.
In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of Kähler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local collapsing estimate and can be regarded as an analogue of the Bishop-Gromov volume comparison for Ricci flow.
The Canonical Expanding Soliton and Harnack inequalities for Ricci flow
引入正则扩展Ricci孤子的概念，并利用它导出了里奇流的新的Harnack不等式.这一观点也给出了几何洞察到现有的Harnack不等式的Hamilton和Brendle。
Convergence Results for Two Kahler-Ricci Flows
本文对K的研究中的两个主要版本进行了讨论闭流形上的里奇流，目的是光滑收敛到相应的Kahler-Einstein度规，并假设了沿流的体积形式和里奇曲率形式。
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