John W. Morgan, Gang Tian
This manuscript contains a detailed proof of the Poincare Conjecture. The arguments we present here are expanded versions of the ones given by Perelman in his three preprints posted in 2002 and 2003. This is a revised version taking in account the comments of the referees and others. It has been reformatted in the AMS book style.
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In this thesis we give a review on Ricci flow, an overview on Poincare conjecture, maximum principle, Li-Yau-Perelman estimate, Two functional F and W of Perelman, Reduced volume and reduced length and k-non collapsing estimate
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
We show that the solution constructed in an earlier work of Y-G. Shi and the authors can be used to obtain sharp gradient estimates for the Kaehler-Ricci flow which achieves equality on a steady soliton. The estimate can be applied to obtain a long time existence of the Kaehler-Ricci flow. In the second part of the paper we develope a so-called "moment" type estimate for the Kaehler-Ricci flow, through which we prove that the flow preserves several properties. In the last part we study the convergence of the Kaehler-Ricci flow.
This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it depends on the initial total curvature and the initial length, rather than just on the initial total curvature as was asserted before. This change does not affect the application of these results to prove finite-time extinction when the third homotopy group of the manifold is non-tr…
We provide here a counter-example to the second inequality of Corollary (19.10) in the Clay Institute Monograph by J.Morgan and G.Tian entitled "Ricci Flow and the Poincare Conjecture". We had announced the existence of this counter-example in our paper "Five Gaps in Mathematics", Advanced Non-linear Studies, vol 15, No. 2, (2015). We make the details available here. J.Morgan and G.Tian have recently (arXiv/math/DG:1512.00699, (2015)) published a correction to their arguments in the monograph. In a forthcoming short Note, we will discuss this correction. We wish to thank John Morgan, Terry Tao…
In [Cheeger-Tian 2005], Cheeger-Tian proved an $ε$-regularity theorem for $4$-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensional manifolds and higher dimensional Einstein manifolds. In this paper we consider all these problems. First, we construct counterexamples to the conjecture for $4$-dimensional critical metrics and counterexamples to the conjecture for higher dimensional Einstein manifolds. For $4$-dimensional shrinking Ric…
As a consequence of the Bochner formula for the Bismut connection acting on gradients, we show sharp universal Poincaré and log-Sobolev inequalities along solutions to generalized Ricci flow. Using the two-form potential we define a twisted connection on spacetime which determines an adapted Brownian motion on the frame bundle, yielding an adapted Malliavin gradient on path space. We show a Bochner formula for this operator, leading to characterizations of generalized Ricci flow in terms of universal Poincaré and log-Sobolev type inequalities for the associated Malliavin gradient and Ornstein-…
We introduce the notion of contact Ricci flow associated with the Reeb vector field. Using it, we give a simple proof of the Poincare conjecture.
Combinatorial Ricci Flow and Thurston's Triangulation Conjecture
本文提出组合Ricci流作为处理Thurston三角剖分猜想的一般方法，证明流收敛当且仅当三角剖分是几何的，并建立了一般双曲多面体3-流形的刚性结果。
We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conje…
Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincaré inequality without assuming the Ricci curvature is bounded from below.
In this paper, we study the behavior of Bergman kernels along the Kähler Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler Ricci flow under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture of Tian for Fano manifolds of complex dimension $\le 3$.
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in \cite{Bau2}. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster-Tanaka-Ricci curvature is non negative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is …
In this article we study the limiting behavior of the Kähler Ricci flow on complete non-compact Kähler manifolds. We provide sufficient conditions under which a complete non-compact gradient Kähler-Ricci soliton is biholomorphic to $\ce^n$. We also discuss the uniformization conjecture by Yau \cite{Y} for complete non-compact Kähler manifolds with positive holomorphic bisectional curvature.
We discuss some of the key ideas of Perelman's proof of Poincaré's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.
We consider the Ricci flow on $\mathbb{CP}^n$ blown-up at one point starting with any $U(n)$-invariant Kähler metric. It is known that the Kähler-Ricci flow must develop Type I singularities. We show that if the total volume does not go to zero at the singular time, then any Type I parabolic blow-up limit of the Ricci flow along the exceptional divisor is the unique $U(n)$-complete shrinking Kähler-Ricci soliton on $\mathbb C^n$ blown-up at one point. This establishes the conjecture of Feldman-Ilmanen-Knopf.
In this short note we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about next steps.
We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds $V_2\mathbb{R}^n$ and $V_{1+k_2}\mathbb{R}^n$, with $n = 1+k_2+k_3$. We use techniques from the theory of differential equations, in particular the Poincaré compactification.
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.
The aim of this paper is to give a proof the Frankel conjecture by using the Kahler Ricci flow alone without assuming apriori the existence of Kahler Einstein metrics. However, there is an essential difference between the real case and the Kahler case. I didn't realize this difference in the calculation of the previous version thus made a mistake on the crucial point.
This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; however, in a homogeneous manifold, the Ricci Flow reduces to an ODE. The behavior of Ricci Flow in two, three, and four dimensional homogenous manifolds has been calculated and is well understood. The work presented here will describe efforts to better understand the behavior of R…
In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.
Yau's uniformization conjecture states: a complete noncompact Kähler manifold with positive holomorphic bisectional curvature is biholomorphic to $\ce^n$. The Kähler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the Kähler-Ricci flow with focus on its application to uniformization. Other interesting methods and results related to the study of Yau's conjecture are also discussed.
Verifying a conjecture of Gromov we establish a generalized Margulis Lemma for manifolds with lower Ricci curvature bound. Among the various applications are finiteness results for fundamental groups of compact $n$-manifolds with upper diameter and lower Ricci curvature bound modulo nilpotent normal subgroups.
A Lean Formalization of the Hamilton--Perelman Proof of the Three-Dimensional Poincaré Conjecture
本文用 Lean 形式化证明了三维庞加莱猜想，结合 Moise 定理，采用 Ricci 流与手术及有限时间消没的 Hamilton–Perelman 方法。
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.
On $κ$-solutions and canonical neighborhoods in 4d Ricci flow
本研究提出四维Ricci流中$\kappa$-解的分类猜想，构造了新的对称单参数族解，建立了穿过柱形奇点的典范邻域定理，并指出商颈带来的新现象与非唯一性示例。
In this short note, we show that homogeneous Ricci solitons are algebraic. As an application, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.
In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show that the conjecture is true under an additional condition. Furthermore, such a soliton must be of constant curvature.
Cao's splitting theorem says that for any complete Kähler-Ricci flow $(M,g(t))$ with $t\in [0,T)$, $M$ simply connected and nonnegative bounded holomorphic bisectional curvature, $(M,g(t))$ is holomorphically isometric to $\C^k\times (N,h(t))$ where $(N,h(t))$ is a Kahler-Ricci flow with positive Ricci curvature for $t>0$. In this article, we show that $k=n-r$ where $r$ is the Ricci rank of the initial metric. As a corollary, we also confirm a splitting conjecture of Wu-Zheng when curvature is assumed to be bounded.
On Hopf's conjecture and positive second intermediate Ricci curvature
针对正第二中间 Ricci 曲率下 Hopf 猜想结论可能失效的问题，本文证明维数被四整除且具有秩至少十的环面对称性时，Euler 示性数为正，并推广了 Four Periodicity Theorem。
We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons and nilsolitons. We obtain that any homogeneous expanding Ricci soliton M=G/K is diffeomorphic to a product U/K x N, where U is a maximal reductive Lie subgroup of G and N is the maximal nilpotent normal subgroup of G, such tha…
We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3 dimensional ancient $κ$ solutions to the Ricci flow. As an application, using the $W$ entropy associated with the heat kernel, we give a different and shorter proof of Perelman's classification of backward limits of these ancient solutions. The current paper together with \cite{Z:2} and a different proof of universal noncollapsing due to Chen and Zhu \cite{ChZ:1} lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.
In this paper, we will establish a regularity theory for the Kähler-Ricci flow on Fano $n$-manifolds with Ricci curvature bounded in $L^p$-norm for some $p > n$. Using this regularity theory, we will also solve a long-standing conjecture for dimension 3. As an application, we give a new proof of the Yau-Tian-Donaldson conjecture for Fano 3-manifolds. The results have been announced in \cite{TiZh12b}.
Let $(X, D)$ be a log variety with an effective holomorphic torus action, and $Θ$ be a closed positive $(1,1)$-current. For any smooth positive function $g$ defined on the moment polytope of the torus action, we study the Monge-Ampère equations that correspond to generalized and twisted Kähler-Ricci $g$-solitons. We prove a version of Yau-Tian-Donaldson (YTD) conjecture for these general equations, showing that the existence of solutions is always equivalent to an equivariantly uniform $Θ$-twisted $g$-Ding-stability. When $Θ$ is a current associated to a torus invariant linear system, we furth…
In this article we classify expanding homogeneous Ricci solitons up to dimension 5, according to their presentation as homogeneous spaces. We obtain that they are all isometric to solvsolitons, and this in particular implies that the generalized Alekseevskii conjecture holds in these dimensions. In addition, we prove that the conjecture holds in dimension 6 provided the transitive group is not semisimple.
This is largely an exposition article that expands the author's talk at the Xiamen International Conference on Geometric Analysis in June 2021. We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD) conjecture for $g$-weighted Kähler-Ricci solitons ($g$-solitons). We then review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a connection between a particular $g$-soliton equation with Ricci-flat Kähler cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in this connection is the transformation of a possibly irregular Sa…
Based on the compactness of the moduli of non-collapsed Calabi-Yau spaces with mild singularities, we set up a structure theory for polarized Kähler Ricci flows with proper geometric bounds. Our theory is a generalization of the structure theory of non-collapsed Kähler Einstein manifolds. As applications, we prove the Hamilton-Tian conjecture and the partial-$C^0$-conjecture of Tian.
Kähler-Ricci shrinkers and Fano fibrations
本文用双有理代数几何证明Kähler-Ricci收缩孤立子是拟射影簇，并提出将其存在性与极化Fano纤维化K-稳定性及奇点切流相联系的统一猜想。
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 curvature and $\ep>0$ is a uniform constant, $M^3$ is compact. One of the key ingredients to exclude the local collapse in singularities of the Ricci flow is the use of pinching-decaying estimate. The other important part of our argument is to role out the Type III singularity complete noncompact Ricci flow with positive Ricci pinching condition. We get thi…
The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. This article confirms this conjecture. As a special case, this gives a new method of uniformization of the Riemann sphere.
A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound
本文在 Ricci 曲率下界条件下证明了小体积情形的广义 Cartan-Hadamard 猜想，并给出任意体积情形向等式刚性归约的关键简化。
Scalar-flat Kähler surfaces whose Weyl tensor annihilates the Ricci form
研究标量平坦凯勒表面中威利张量作用于2形式 annihilates 里奇形式的性质，证明其为里奇平坦或局部等距于两个曲率相反的实表面乘积。
The Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential invariants and derive new formulae for several other classification problems in geometry and analysis.
This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $M$ is bounded for $H\ne 1$. In [CRX], we verified the conjecture for the case that $M$ or its Riemannian universal covering space $\tilde M$ is not collapsed for $H=1$ o…
We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelman's conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds --- i…
Laplace comparison on Kähler Ricci flow and convergence
该研究证明Fano流形上Kähler Ricci流的一致积分Laplace比较结果，结合相关理论与已有成果，给出Hamilton-Tian猜想的直接独立证明，并梳理相关文献。
We study the evolution of anticanonical line bundles along the Kähler Ricci flow. We show that under some conditions, the convergence of Kähler Ricci flow is determined by the properties of the anticanonical divisors of $M$. As examples, the Kähler Ricci flow on $M$ converges when $M$ is a Fano surface and $c_1^2(M)=1$ or $c_1^2(M)=3$. Combined with the work in \cite{CW1} and \cite{CW2}, this gives a Ricci flow proof of the Calabi conjecture on Fano surfaces with reductive automorphism groups. The original proof of this conjecture is due to Gang Tian.
Moving Manifolds and the Poincare Conjecture
本文用移动曲面微积分给出庞加莱猜想的微分几何证明框架，通过变分原理使紧致单连通超曲面演化至常平均曲率平衡态，最终收敛于球面，统一了动力学、拓扑与几何。
We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and Hsiang-Kleiner theorem for positively curved manifolds do not hold in the $\mathrm{Ric}_2>0$ setting. These observations indicate that the class of manifolds with $\mathrm{Ric}_2>0$ is vastly different from the class of positively curved manifolds.
In this paper, we prove the Hamilton-Tian conjecture for Kähler-Ricci flow based on a recent work of Liu-Székelyhidi on Tian's partical $C^0$-estimate for poralized Kähler metrics with Ricci bounded below. The Yau-Tian-Donaldson conjecture for the existence of Kähler-Einstein metrics on Fano manifolds will be also discussed.
This memoire consists of two main results. In the first one we describe Ricci flow theory and we give an educative way for proving Elliptization Conjecture and then we prove Poincare conjecture which is the second proof of Perelman for Poincare conjecture. In the second one which is the main propose of our memoire, we exhibit a complete proof of Calabi-Yau conjecture.
In this paper we study the behavior of the Ricci flow at infinity for the full flag manifold $SU(3)/T$ using techniques of the qualitative theory of differential equations, in special the Poincaré Compactification and Lyapunov exponents. We prove that there are four invariant lines for the Ricci flow equation, each one associated with a singularity corresponding to a Einstein metric. In such manifold, the bi-invariant normal metric is Einstein. Moreover, around each invariant line there is a cylinder of initial conditions such that the limit metric under the Ricci flow is the corresponding Ein…
The intent of this short note is to provide context for and an independent proof of the discovery of Klaus Kroencke that complex projective space with its canonical Fubini--Study metric is dynamically unstable under Ricci flow in all complex dimensions N>1. The unstable perturbation is not Kaehler. This provides a counterexample to a well known conjecture widely attributed to Hamilton. Moreover, it shows that the expected stability of the subspace of Kaehler metrics under Ricci flow, another conjecture believed by several experts, needs to be interpreted in a more nuanced way than some may hav…
In this paper we give an explicit bound of $Δ_{g(t)}u(t)$ and the local curvature estimates for the Ricci-harmonic flow under the condition that the Ricci curvature is bounded along the flow. In the second part these local curvature estimates are extended to a class of generalized Ricci flow, introduced by the author \cite{LY1}, whose stable points give Ricci-flat metrics on a complete manifold, and which is very close to the $(K, N)$-super Ricci flow recently defined by Xiangdong Li and Songzi Li \cite{LL2014}. Next we propose a conjecture for Einstein's scalar field equations motivated by a …
There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.
We study singularity formation of complete Ricci flow solutions, motivated by two applications: (a) improving the understanding of the behavior of the essential blowup sequences of Enders-Muller-Topping on noncompact manifolds, and (b) obtaining further evidence in favor of the conjectured stability of generalized cylinders as Ricci flow singularity models.
We show that noncompact homogeneous spaces not diffeomorphic to Euclidean space of dimension 9 or 10 admit no homogeneous Einstein metrics of negative Ricci curvature, with only three potential exceptions. The main ingredient in the proof is to show, via a cohomogeneity-one approach, that noncompact homogeneous spaces admitting an ideal isomorphic to $ \mathfrak{sl}_2(\mathbb{R}) $ admit no homogeneous Einstein metrics of negative Ricci curvature.
Uniqueness results for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary
本文研究了非负里奇曲率且严格凸边界的紧黎曼流形上正调和函数的Liouville型定理，验证了Wang猜想的部分情况，并通过Warped积流形进一步探讨了该猜想。
On the Hamilton-Tian Conjecture in a compact transverse Fano Sasakian $5$-manifold
本文确认了在紧致横贯Fano准正则萨克森5流形中Sasaki-里奇流的Hamilton-天猜想，并推导了横贯Fano准正则萨克森5流形上Sasaki-里奇溶洞的紧致性定理，进一步通过第二萨克森结构定理确认了紧致横贯Fano萨克森5流形中的Hamilton-天猜想，并展示了梯度Sasaki-里奇溶洞轨道度量在紧致萨克森5流形上为Sasaki-爱因斯坦的条件。
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