Brett Kotschwar
We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution which converges modulo diffeomorphisms to a soliton faster than any fixed exponential rate must itself be self-similar.
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Type II extinction profile of maximal solutions to the Ricci flow in $\R^2$
本文考虑了$u_t = Δ\log u$，$u(x,0) = u_0(x)\ge 0$在$\R^2$中的初值问题，对应于里奇流，即度量$u (dx_1^2 + dx_2^2)$通过Ricci曲率的共形演化.众所周知，最大（完全）解$u$在时间$T= \frac 1{4π} \int_{\R^2} u_0 $之后相同地消失。假设$u_0$是compatible支持的，我们精确地描述了$u$在时间$T$的II型消失行为：我们显示了存在一个指数快速消失的轮廓，这是一个内部区域，适当的缩放，一个{\em孤子雪茄解决方案}，和存在一个对数尖点的持久性的外部区域。这是迄今为止，在任何维的里奇流中唯一被证明存在的II型奇点。它恢复了严格的形式渐近由J.R.国王\cite{K}。
Almost-rigidity and the extinction time of positively curved Ricci flows
证明了几乎具有最大消亡时的里奇流在与$\mathbb{R}^{2}$相交时具有正的迷向曲率的条件下一定是近圆流.作为应用，我们证明了$S^{3}$和$RP^{3}$上具有几乎最大宽度的正曲度量必须是近圆的.
On the Perelman's reduced entropy and Ricci flat manifolds with maximal volume growth
本文利用Perelman的里奇流的约化体积研究了具有最大体积增长的Ricci平坦流形。证明了若$(M^n,g)$是一个非紧完备Ricci平坦流形，其最大体积增长满足$|Rm|(x)\to 0$为$d(x)=d_g(x,p)\to \infty$，则$M^n$具有二次曲率衰减.最后给出了这一结果的一些应用。
Convergence of Ricci flow on $R^2$ to plane
本文给出了$R^2$中的里奇流全局存在且在$t=\infty$收敛到$R^2$上平坦度量的充分条件.
An elementary proof of the convergence of Ricci flow on compact surfaces
这篇论文已被作者撤回，以便进一步修改。
Let $M$ be an open Riemannian $n$-manifold with nonnegative Ricci curvature. We prove that if the first Betti number of $M$ equals $n-1$, then $M$ is flat.
In a previous work, the authors have introduced a Lin-Lu-Yau type Ricci curvature for directed graphs, and obtained a diameter comparison of Bonnet-Myers type. In this paper, we investigate rigidity properties for the equality case, and conclude a maximal diameter theorem of Cheng type.
On stability and the convergence of the Kähler-Ricci flow
假设曲率有一致的界，Kähler-里奇流的指数收敛在两个条件下成立，这两个条件是稳定性的一种形式：Mabuchi能量是下有界的，全纯向量场空间的维数在全纯群的轨道上不能在极限上跳起来.
Convergence of Ricci flow on $\mathbb{R}^2$ to flat space
我们证明了，从一个初始度量$g(0)=e^{2u_0}(dx^2+dy^2)$在$\mathbb{R}^2$有界的数量曲率和有界的$u_0$，里奇流$\partial_t g(t)=-R_{g(t)}g(t)$收敛到一个平坦的度量在$\mathbb{R}^2$。
This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to addre…
A general convergence result for the Ricci flow in higher dimensions
设（M，g_0）是n维紧致黎曼流形\geq 4.我们证明了如果（M，g_0）x R具有正的各向同性曲率，则归一化的里奇流使g_0变形为常曲率度量.这个条件比2-正旗曲率算子强，但比2-正曲率算子弱。
In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away fro…
We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to $\mathbb{R}^n$.
Positive intermediate Ricci curvature with maximal symmetry rank
该研究推广了Grove和Searle的工作，证明了正中间里奇曲率流形等距群秩的上界，并在最大对称秩情形下获得更强的拓扑刚性结果，涵盖更高曲率及非平凡基本群。
In this short note we prove that a Kahler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results of Kewei Zhang and Yuchen Liu on holomorphic rigidity of such Kahler manifolds with the structure theorem of Tian-Wang for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume
Stability of Kähler-Ricci flow
证明了Kähler-里奇流在某些小初曲率条件下的收敛性。作为应用，我们讨论了Kähler-Einstein流形上复结构变化时Kähler-里奇流的收敛性.
Convergence and stability of locally \mathbb{R}^{N}-invariant solutions of Ricci flow
里奇流的不朽解的有界曲率坍塌的重要模型来自于主丛上的局部G-不变解，其中G是一个幂零的李群。在本文中，我们建立了某些R^{N}-不变解的收敛性和渐近稳定性，模光滑有限维中心流形。当全空间维数为3时，这些结果与Lott对截面曲率和直径分别为O（t^{-1}）和O（t^{1/2}）的所有三维里奇流解的渐近行为进行分类的工作有关.
We study the problem of convergence of the normalized Ricci flow evolving on a compact manifold $Ω$ without boundary. In \cite{KS10, KS15} we derived, via PDE techniques, global-in-time existence of the classical solution and pre-compactness of the orbit. In this work we show its convergence to steady-states, using a gradient inequality of Łojasiewicz type. We have thus an alternative proof of \cite{ha}, but for general manifold $Ω$ and not only for unit sphere. As a byproduct of that approach we also derive the rate of convergence according to this steady-sate being either degenerate or non-d…
Kaehler manifolds with numerically effective Ricci class and maximal first Betti number are tori
设M是一个具有数值有效Ricci类的n维Kähler流形。本文证明了：若第一个贝蒂数b_1（M）=2n，则M与复环面T^n_C双全纯.
The principle of convergence stability for geometric flows is the combination of the continuous dependence of the flow on initial conditions, with the stability of fixed points. It implies that if the flow from an initial state $g_0$ exists for all time and converges to a stable fixed point, then the flows of solutions that start near $g_0$ also converge to fixed points. We show this in the case of the Ricci flow, carefully proving the continuous dependence on initial conditions. Symmetry assumptions on initial geometries are often made to simplify geometric flow equations. As an application o…
Convergence of the Ricci flow to Ricci-flat ALE manifolds and positive scalar curvature rigidity
研究里奇流下具有平行旋量的可积ALE流形稳定性，给定特定初始度量，结果适用于4维引力瞬子，衰减率能证明\(L^p\)中正标量曲率刚性，推广了阿普尔顿的结果。
Convergence of the Kähler-Ricci iteration
Ricci迭代是里奇流的离散模拟。根据佩雷尔曼的说法，只要存在一个卡勒-爱因斯坦度规，里奇流就收敛到卡勒-爱因斯坦度规，并且已经证明里奇迭代应该表现出类似的行为。本文证实了这一猜想。作为一种特殊情况，这给出了一种新的黎曼球面的单值化方法。
The volume entropy of a compact metric measure space is known to be the exponential growth rate of the measure lifted to its universal cover at infinity. For a compact Riemannian $n$-manifold with a negative lower Ricci curvature bound and a upper diameter bound, it was known that it admits an almost maximal volume entropy if and only if it is diffeomorphic and Gromov-Hausdorff close to a hyperbolic space form. We prove the quantitative rigidity of almost maximal volume entropy for $\operatorname{RCD}$-spaces with a negative lower Ricci curvature bound and Riemannian manifolds with a negative …
Convergence of Ricci flow and long-time existence of Harmonic map heat flow
本文建立了渐近紧致可积shrinker的ancient Ricci flow及相应奇点情形下harmonic map heat flow的长时间存在性，提供全局parabolic gauge并证明切流唯一性，同时分类了ancient flows并得出奇点最优收敛速率。
Convergence stability for Ricci flow on manifolds with bounded geometry
本文证明有界几何完备度量下Ricci流在有限时间内连续依赖初值且不损失正则性，并借助Ricci-DeTurck流生成元的扇形性，证明接近旋转对称渐近双曲度量的初始度量在适当曲率条件下Ricci流收敛到双曲度量。
A finite Topological Type Theorem for open manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume
针对非负 Ricci 曲率且局部重绕体积近最大的开流形，提出无需截面曲率、共轭半径或 Toponogov 型比较假设的有限拓扑类型定理，并扩展了既有方法的适用范围。
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.
Kähler-Ricci solitons with almost maximal symmetry
本文研究高对称性的非平凡梯度Kähler-Ricci孤立子，证明其等距群为一齐性作用且可采用Sasakian模型拟设，复二维时必具最大对称性，还将势函数不变性推广到一般梯度Ricci孤立子情形。
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。
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 $δ(n, d, p), ε(n, d, p)>0$, such that for $δ<δ(n, d, p)$, $ε<ε(n, d, p)$, if a compact $n$-manifold $M$ satisfies that the integral Ricci curvature has lower bound $\bar k(-1, p)\leq δ$, the diameter $diam(M)\leq d$ and volume entropy $h(M)\geq n-1-ε$, then the universal cover of $M$ is Gromov-Hausdorff close to a hyperbolic space form $\Bbb H^k$, $k\leq n$; If in addition the volume of $M$, $vol(M)\geq…
We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric $g_{0}$ on $\mathbb{R}^{4}$ with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If $g_{0}$ has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric $g_{\mathsf{TNUT}}$ in the Cheeger-Gromov sense in infinite time. We also classify the long-time behaviour when $g_{0}$ is asymptotically flat. In order to identify infinite-time singularity model…
Singular Ricci Flows on surfaces with boundary and positive scalar curvature
本文研究带边界紧致曲面上满足给定常值空间测地曲率的奇异 Ricci 流解的子序列收敛，并证明旋转对称情形下该收敛与边界测地曲率符号无关。
Existence of Ricci flows of incomplete surfaces
本文证明了起始于任意黎曼曲面的瞬时完全里奇流的一般存在性结果，该黎曼曲面可能是不完全的且可能具有无界曲率。我们给出了一个显式公式的最大存在时间，并描述了在大多数情况下的渐近行为。
Quantitative Maximal Diameter Rigidity of Positive Ricci Curvature
本文证明在正 Ricci 曲率下，若流形直径接近最大直径且局部万有覆盖满足 Riefenberg 条件，则其与单位球面微分同胚且双 Hölder 接近，从而将 Cheng 刚性定理定量化。
Einstein solvmanifolds have maximal symmetry
所有已知的负里奇曲率的齐次爱因斯坦度量的例子都可以在可解李群上实现为左不变黎曼度量。在李群上定义了左不变黎曼度量之间的极大对称性概念后，证明了可解李群上任何具有负里奇曲率的左不变Einstein度量都是极大对称的.这个定理是由Alekseevskii猜想和爱因斯坦度量在里奇流下的稳定性问题激发的。我们还解决了最大对称左不变黎曼度量的存在性问题。
On the convergence of the Sasaki-Ricci flow
给定Sasaki流形S，证明了Sasaki-里奇流指数快速收敛到Sasaki-Einstein度量，如果存在的话，假设横截全纯结构的自同构群是平凡的.
Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)"
该论文针对Kähler Ricci流相关研究，明确了极化典范半径的定义，补充完善了相关概念的界定。
Convergence of Kähler-Ricci flow on Fano manifolds, II
本文给出了Fano流形$(M,J)$上Kähler-里奇流收敛性的另一种证明.这个证明不同于[TZ 3]中的证明。此外，我们将[TZ 3]的主要定理推广到$(M,J)$不允许任何Kähler-Einstein度量的情形.
In this paper, we study the uniformly strong convergence of Kähler-Ricci flow on a Fano manifold with varied initial metrics and smooth deformation complex structures. As an application, we prove the uniqueness of Kähler-Ricci solitons in sense of diffeomorphism orbits. The result generalizes Tian-Zhu's theorem for the uniqueness of of Kähler-Ricci solitons on a compact complex manifold, and it is also a generalization of Chen-Sun's result of for the uniqueness of of Kähler-Einstein metric orbits.
Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces
研究了具有非退化Poisson结构的复曲面上的广义Kähler-里奇流，证明了该流的长时间存在性和收敛于弱超Kähler结构.
Convergence of Finslerian metrics under Ricci flow
本文研究了在一般流中演化的Finslerian度量在Finslerian里奇流下的收敛性。更直观地证明了当$ t $逼近有限时间$ T $时，作为Finslerian里奇流解的Finslerian度量族$g(t)$在$C^{\infty}$中收敛到一个光滑极限Finslerian度量.作为这个结果的一个结果，我们可以证明，在紧致Finsler流形中，沿里奇流的曲率张量在短时间内爆破。
Convergence of a Kähler-Ricci flow
本文证明了：对于任意维数上具有一致有界里奇曲率的Kähler-里奇流，对于每一个收敛到无穷大的时间序列$t_i$，存在一个子序列使得$(M,g(t_i + t))\to (Y,\bar{g}(t))$，并且在一个奇异集（余维数至少为4的集合）外，收敛到流的解是光滑的.我们还证明了在复维数为2的情况下，没有任何曲率假设，我们可以找到一个子序列的时间，使我们有一个收敛到一个Kähler-Ricci孤子，远离许多孤立奇点。
We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.
We study the quasi-convergence equivalence of some families of metrics on locally homogeneous closed 4-manifolds with trivial isotropy group, and identify the dimension of each equivalence class under certain conditions.
We extend some convergence results on nonsingular compact Ricci flows in the papers \cite{Ha:1}, \cite{Se:1} and \cite{FZZ:2} to certain infinite volume noncompact cases which are "partially" nonsingular. As an application, for a finite time singularity which is partially type I, it is shown that a blow up limit is a gradient shrinking soliton.
The Backward behavior of the Ricci and Cross Curvature Flows on SL(2,R)
本文研究SL（2，R）型局部齐次三流形上的Ricci流和交叉曲率流的极大解的性质。我们证明，一般来说，一个极大解起源于一个海森堡型的次黎曼几何。这解决了两个作者在早期工作中留下的一个问题。
Convergence of Kähler-Ricci flow on lower dimensional algebraic manifolds of general type
本文证明了里奇曲率的$L^4$ -范数沿着任意极小代数流形上的Kähler-里奇流是一致有界的.作为应用，我们证明了在任意一般型的极小代数流形$M$上，在Cheeger-Gromov拓扑下的正则模型$M$上，正规化Kähler-里奇流的任意解收敛于唯一奇异Kähler-Einstein度量.
On convergence of the Kähler-Ricci flow
研究了Fano流形上Kähler-里奇流在一定稳定性条件下的收敛性.更精确地说，我们假设作用于向量场的$\bar\partial$-算子的第一特征值沿流是一致有界的，此外，马渊能量至多以几何方式衰减。然后，我们给出不同的情况下，条件的马渊能量举行。
A caveat on the convergence of the Ricci flow for pinched negatively curved manifolds
我们给出的例子，捏负曲流形的里奇流不收敛顺利。
Intrinsic flat and Gromov-Hausdorff convergence of manifolds with Ricci curvature bounded below
我们证明了对于一个非坍缩的闭连通定向黎曼流形序列，其里奇曲率从下到上一致有界，直径从上到下一致有界，格罗莫夫-豪斯多夫收敛本质上与内在平坦收敛一致.
We study the generalized Kähler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We derive an extension of Perelman's entropy functional to this setting, which yields convergence of nonsingular solutions at infinity. Furthermore, we derive an extension of Mabuchi's $K$-energy to this setting, which yields weak convergence of the flow.
On the Convergence of a Modified Kaehler-Ricci flow
研究了Zhou Zhang定义的一个修正的Kaeher-里奇流的收敛性.我们证明了当极限类退化时，流收敛到一个奇异度量。这证明了张的一个猜想。
In this paper, we consider Kahler-Ricci flow on n-dimensional Kahler manifold with semi-ample canonical line bundle and 0< m:= Kod(X)<n. Such manifolds admit a Calabi-Yau fibration over its canonical model. We prove that the scalar curvature of the Kahler metric along the normalized Kahler-Ricci flow converge to -m outside the singular set of this fibration.
Convergence of compact Ricci solitons
我们证明了在体积、曲率的$L^{n/2}$ -范数和辅助常数$C_1$的约束下，紧致梯度Ricci孤子序列收敛到完整的轨道折叠梯度孤子.最强的结果是在4维，其中$L^2$曲率边界是等价的欧拉数的上界。得到了极限是紧极限的充要条件。
We give a complete description of the global existence and convergence for the Ricci-Yang-Mills flow on $T^k$ bundles over Riemann surfaces. These results equivalently describe solutions to generalized Ricci flow and pluriclosed flow with symmetry.
Chern-Ricci flows on noncompact complex manifolds
本文得到了非紧流形上Chern-里奇流的存在性准则。本文将Tossati-Wienkove关于Chern-里奇流的一个结果推广到非紧流形，同时将Lott-Zhang关于Kahler-里奇流的一个结果推广到Chern-里奇流.利用存在性结果，证明了非紧复流形上任何具有非负二分曲率的完备非塌陷Kahler度量都可以变形为具有非负有界二分曲率的完备Kahler度量，且当初始度量具有最大体积时，该度量具有最大体积增长.结合这个结果和Chau-Tam的结果，我们给出了具有非负双截曲率（不一定有界）和最大体积增长的完备非紧Kahler流形与复欧几里得空间双全纯的另一个证明.这最后一个结果已经证明了刘刚最近使用其他方法。这最后一个结果部分证实了丘的单值化猜想。
Volume estimates and convergence results for solutions to Ricci flow with $L^{p}$ bounded scalar curvature
本文研究具有 $L^p$ 有界数量曲率的 Ricci 流，证明四维闭流形非塌缩估计，结合积分界获得改进的 Ricci 曲率时空积分界，并证明解收敛到 orbifold 且可延拓。
On the long time behaviour of the Conical Kähler- Ricci flows
本文证明了在\cite{CYW}中引入的圆锥Kähler-里奇流在$t\in [0,+\infty)$中的任何时刻都存在.这些不朽的流在圆锥范畴中具有最大的正则性。作为应用，我们证明了当扭曲的第一类陈氏$C_{1,β}$为负或为零时，相应的圆锥Kähler-里奇流以指数速度收敛到具有圆锥奇点的Kähler-Einstein度量.为了建立这些结果，我们的关键步骤之一是证明具有圆锥奇点的Kähler-Ricci平坦度量（定义在$\mathbb{C}^{n}$上）的Liouville型定理。
The transverse Chern-Ricci flow
本文对横向厄米叶理引入了类似于Chern-里奇流的横向Chern-里奇流。我们表明，当$\mathcal{F}$是同调定向和基本的第一Bott-Chern类为零，从任何横向厄米度量的流量存在的所有时间和$t\rightarrow \infty$顺利收敛到一个横向厄米度量$ω_\infty$与横向陈里奇形式$ρ^T(ω_\infty)=0$。在一般情况下，我们还确定了流的最大存在时间。这些结果是Gill和Tosatti-Weinkove的结果的叶状版本，也是Bedulli-He-Vezzoni最近工作的推广。
In this article we use Ricci flow to show that complete PIC1 manifolds with maximal volume growth are diffeomorphic to $\mathbb{R}^n$. One of the key ingredients is local estimates of curvature lower bounds on an initial time interval of the Ricci flow. As another application of these estimates we obtain pseudolocality type results related to the PIC1 condition.
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