A Survey of Hamilton's Program for the Ricci Flow on 3-manifolds

Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature

In this paper we derive Cheng-Yau, Li-Yau, Hamilton estimates for Riemannian manifolds with Bakry-Emery Ricci curvature bounded from below, and also global and local upper bounds, in terms of Bakry-Emery Ricci curvature, for the Hessian of positive and bounded solutions of the weighted heat equation on a closed Riemannian manifold.

Eigen-value monotonicity for the Ricci-Hamilton flow

This paper has been withdrawn by the author and it is published in AGAG

Hamilton-Souplet-Zhang's gradient estimates for two types of nonlinear parabolic equations under the Ricci flow

In this paper, we consider gradient estimates for two type of nonlinear parabolic equations under the Ricci flow: one is the equation $$u_t=Δu+au\log u+bu$$ with $a,b$ two real constants, the other is $$u_t=Δu+λu^α$$ with $λ,α$ two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-Souplet-Zhang type gradient estimates.

A generalization of Hamilton's differential Harnack inequality for the Ricci flow

In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature.

Remarks on Hamilton's Compactness Theorem for Ricci flow

A fundamental tool in the analysis of Ricci flow is a compactness result of Hamilton in the spirit of the work of Cheeger, Gromov and others. Roughly speaking it allows one to take a sequence of Ricci flows with uniformly bounded curvature and uniformly controlled injectivity radius, and extract a subsequence that converges to a complete limiting Ricci flow. A widely quoted extension of this result allows the curvature to be bounded uniformly only in a local sense. However, in this note we give a counterexample.

Kähler-Ricci流下的矩阵Li-Yau-Hamilton估计

Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow

该研究证明了Kähler-Ricci流耦合下热方程与反向共轭热方程正解的矩阵Li-Yau-Hamilton估计，并推导得到单调性公式，丰富了几何分析相关理论。

Equivalence of Simplicial Ricci Flow and Hamilton's Ricci Flow for 3D Neckpinch Geometries

Hamilton's Ricci flow (RF) equations were recently expressed in terms of the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry, for d greater than or equal to 2. The structure of the simplicial Ricci flow (SRF) equations are dimensionally agnostic. These SRF equations were tested numerically and analytically in 3D for simple models and reproduced qualitatively the solution of continuum RF equations including a Type-1 neckpinch singularity. Here we examine a continuum limit of the SRF equations for 3D neck pinch geometries with an arbitrary radial profile. We show that t…

Ricci-flat Kähler流形中的Hamiltonian stationary Lagrangian流

Hamiltonian stationary Lagrangian currents in Ricci-flat Kähler manifolds

本文研究Ricci-flat Kähler流形中具有单值相位的Hamiltonian stationary Lagrangian流，建立了相位与平均曲率的关系，推导了几乎单调性公式、Allard型正则性定理及极限结构，并应用于复欧几里得空间中该流的正则性与刚性。

Monotonicity and Kaehler-Ricci flow

We show a connection between the linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow and the monotonicity formula for the positive currents. As an application of the linear trace Li-Yau-Hamilton stated in this paper and the one proved by Chow-Hamiltonm, we give another proof on the classification of the limit solutions of the Kaehler-Ricci (Ricci) flow.

Combinatorial Ricci Flows on Surfaces

We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algorithm to find circle packings.

Hamilton-Ivey estimates for gradient Ricci solitons

We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy $|Rm|\leq cR$ for some positive constant $c$. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for $4$-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a $3$-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. …

On an alternate proof of Hamilton's matrix Harnack inequality for the Ricci flow

Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the bundle of 2-forms Whitney sum 1-forms is nonnegative. The idea is to consider the 2-form which minimizes the associated quadratic form to obtain a symmetric 2-tensor. A long but straightforward computation implies this 2-tensor is a subsolution to heat-type equation. A standard a…

Eigenvalue monotonicity for the Ricci-Hamilton flow, revised version

This is a revised version of our short note [arxiv.math.DG/0403065] where we discuss the monotonicity of the eigen-values of the Laplacian operator to the Ricci-Hamilton flow on a compact or a complete non-compact Riemannian manifold. We show that the eigenvalue of the Lapacian operator on a compact domain associated with the evolving Ricci flow is non-decreasing provided the scalar curvature having a non-negative lower bound and Einstein tensor being not too negative. This result will be useful in the study of blow-up models of the Ricci-Hamilton flow. We remark that using our idea, D.Kokotov…

New Li--Yau--Hamilton Inequalities for the Ricci Flow via the Space-time Approach

We generalize Hamilton's matrix Li-Yau-type Harnack estimate for the Ricci flow by considering the space of all LYH (Li-Yau-Hamilton) quadratics that arise as curvature tensors of space-time connections satisfying the Ricci flow with respect to the natural space-time degenerate metric. As a special case, we employ scaling arguments to derive a linear-type matrix LYH estimate. The new LYH quadratics obtained in this way are associated to the system of the Ricci flow coupled to a 1-form and a 2-form evolving by heat-type equations. In the case of a Kaehler solution, a special case of our linear-…

Ricci流与抛物频率下的矩阵Li-Yau-Hamilton估计

Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

该研究证明了与Ricci流耦合的热方程及反向共轭热方程正解的矩阵Li-Yau-Hamilton估计，借此确立带修正因子的抛物频率单调性，并得到曲率非负条件下的唯一延拓结果。

The Wasserstein geometry of non-linear sigma models and the Hamilton-Perelman Ricci flow

Non linear sigma models are quantum field theories describing, in the large deviations sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton-Perelman Ricci flow. By exploiting the heat kernel embedding introduced by N. Gigli and C. Mantegazza, we show that the Wasserstein geometry of the space of probability measures over Riemannian metric measure spaces provides a natural setting for discussing the relation between non-linear sigma m…

非线性势理论与曲率pinched的3维流形

Nonlinear potential theory and Ricci-pinched 3-manifolds

本文通过非线性势理论证明了满足曲率pinching条件和超二次体积增长的三维流形为平坦流形。

A complete proof of Hamilton's conjecture

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…

From the Ricci flow evolution equation to vanishing theorems for Ricci solitons

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.

渐近锥形梯度扩展里奇溶剂的曲率捏紧

Curvature pinching of asymptotically conical gradient expanding Ricci solitons

本文研究了完整非紧致渐近锥形梯度扩展里奇溶剂的曲率捏紧现象，建立了若干Hamilton-Ivey型曲率捏紧估计，包括三维梯度里奇扩展体的正截面曲率结果。

汉密尔顿三维流形定理的精简形式化

A Lean Formalization of Hamilton's Three-Manifold Theorem

该研究完成了汉密尔顿三维流形定理的Lean形式化，采用替代爆破路径而非原始归一化流证明，形式化了里奇流相关关键基础设施及配套流程，为三维流形的里奇曲率研究提供了形式化验证支持。

Aronson-Bénilan estimates for the fast diffusion equation under the Ricci flow

We study the fast diffusion equation (FDE) with a linear forcing term under the Ricci flow on complete manifolds with bounded curvature and nonnegative curvature operator. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of the FDE. In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting.

Kähler-Ricci Flow and the Minimal Model Program for Projective Varieties

In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divisorial contractions for varieties of general type. In case one has a smooth minimal model of general type (i.e., the canonical bundle is nef and big), we show infinite time existence and analyze the singularities.

Sharp differential estimates of Li-Yau-Hamilton type for positive $(p,p)$-forms on Kähler manifolds

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Yau-Hamilton) estimates for the positive solutions of the Hodge-Laplacian heat equation. We also prove a nonlinear version coupled with the Kähler-Ricci flow and some interpolating matrix differential Harnack type estimates for both the Kähler-Ricci flow and the Ricci flow.

The Canonical Expanding Soliton and Harnack inequalities for Ricci flow

We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.

Ricci flow on surfaces along the standard lightcone in the $3+1$-Minkowski spacetime

Identifying any conformally round metric on the $2$-sphere with a unique cross section on the standard lightcone in the $3+1$-Minkowski spacetime, we gain a new perspective on $2d$-Ricci flow on topological spheres. It turns out that in this setting, Ricci flow is equivalent to a null mean curvature flow first studied by Roesch--Scheuer along null hypersurfaces. Exploiting this equivalence, we can translate well-known results from $2d$-Ricci flow first proven by Hamilton into a full classification of the singularity models for null mean curvature flow in the Minkowski lightcone. Conversely, we…

A new Li-Yau-Hamilton estimate for Kahler-Ricci flow

In this paper we prove a new matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate and its linear version proved earlier in \cite{CN}. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian, a manifold version of Stoll's theorem on the …

具有正夹挤Ricci曲率的完备Riemann流形是否是紧的

Is a complete Riemannian manifold with positively pinched Ricci curvature compact

本文针对Hamilton关于欧氏空间中带夹挤第二基本形式的凸超曲面必为紧集的结果，利用作者1998年的拟共形映射结论给出无需额外假设的完整替代证明且可推广，以此纪念Ricci流创立者Hamilton。

Plurisubharmonic functions and the Kaehler-Ricci flow

We prove a linear trace Li-Yau-Hamilton inequality for the Kaehler-Ricci flow. We then use this sharp differential inequality to study the Liouville properties of the plurisubharmonic functions on complete Kaehler manifolds with nonnegative bisectional curvature.

Hamilton恒等式与完备梯度孤立子的刚性

Hamilton's identity and rigidity of complete gradient solitons

本文研究一般几何流的梯度孤立子，提出Hamilton恒等式，并证明该恒等式可恢复关于刚性、势函数增长、体积增长及Omori-Yau最大原理的结果。

Recent Progress on Ricci Solitons

Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton's Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons.

Tian's partial $C^0$-estimate implies Hamilton-Tian's conjecture

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.

Space-time constructions for the mean curvature flow in a Ricci flow background

Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical soliton of the coupled flow within the canonical Ricci soliton. We show that this provides a link between Hamilton's differential Harnack estimate for the mean curvature flow and Hamilton's differential Harnack estimate for the Ricci flow. Moreover the second fundamental form of our…

Bianchi-convex sets and a new maximum principle for the Ricci flow

We introduce the new notion of Bianchi-convex sets, a generalization of convex sets of algebraic curvature tensors inspired by the second Bianchi identity. It turns out that Hamilton's maximum principle for the Ricci flow can be generalized for Bianchi-convex sets.

Compact Gradient Shrinking Ricci Solitons with Positive Curvature Operator

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.

关于紧致横贯Fano萨克森5流形中的Hamilton-天猜想

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。

解析极小模型纲领中的凯勒-里奇切流

Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

该研究针对解析极小模型纲领中的凯勒-里奇流有限时间奇点，在特定假设下证明复二维非坍缩凯勒-里奇流奇点行为的模型，在高维及紧致收缩子情形也获相关结果，完善了相关理论。

Sharp Hamilton's Laplacian estimate for the heat kernel on complete manifolds

In this paper we give Hamilton's Laplacian estimates for the heat equation on complete noncompact manifolds with nonnegative Ricci curvature. As an application, combining Li-Yau's lower and upper bounds of the heat kernel, we give an estimate on Laplacian form of the heat kernel on complete manifolds with nonnegative Ricci curvature that is sharp in the order of time parameter for the heat kernel on the Euclidean space.

Gradient Kähler-Ricci Solitons and Periodic Orbits

We study Hamiltonian dynamics of gradient Kaehler-Ricci solitons that arise as limits of dilations of singularities of the Ricci flow on compact Kaehler manifolds. Our main result is that the underlying spaces of such gradient solitons must be Stein manifolds. Moreover, on all most all energy surfaces of the potential function f of such a soliton, the Hamiltonian vector field V_f of f, with respect to the Kaehler form of the gradient soliton metric, admits a periodic orbit. The latter should be impotant in the study of singularities of the Ricci flow on compact Kaehler manifolds in light of th…

Width and finite extinction time of Ricci flow

This is an expository article with complete proofs intended for a general non-specialist audience. The results are two-fold. First, we discuss a geometric invariant, that we call the width, of a manifold and show how it can be realized as the sum of areas of minimal 2-spheres. For instance, when $M$ is a homotopy 3-sphere, the width is loosely speaking the area of the smallest 2-sphere needed to ``pull over'' $M$. Second, we use this to conclude that Hamilton's Ricci flow becomes extinct in finite time on any homotopy 3-sphere. We have chosen to write this since the results and ideas given her…

Perelman's entropy on ancient Ricci flows

In [ZY2], the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature.

Bartnik's mass and Hamilton's Modified Ricci Flow

We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hamilton's modified Ricci flow and depends only upon the restricted metric of the surface and not on its mean curvature. The theorem is proven by studying a class of asymptotically flat Riemannian manifolds foliated by surfaces satisfying Hamilton's modified Ricci flow with prescribe…

二维空间上稳态梯度Ricci-Yang-Mills孤立子的分类

Classification of Steady Gradient Ricci-Yang-Mills Solitons on Surfaces

本文构造了连接Hamilton cigar与球面的二维弦背景，提出了一族旋转对称的稳态梯度Ricci-Yang-Mills孤立子，并证明所有二维完整稳态梯度Ricci-Yang-Mills孤立子均来自该族。

Ricci flow on locally homogeneous closed 4-manifolds

We discuss the Ricci flow on homogeneous 4-manifolds. After classifying these manifolds, we note that there are families of initial metrics such that we can diagonalize them and the Ricci flow preserves the diagonalization. We analyze the long time behavior of these families. We find that if a solution exists for all time, then the flow exhibits a type III singularity in the sense of Hamilton.

Ricci flow on cone surfaces and a three-dimensional expanding soliton

The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds, introduced by R. Hamilton in 1982. It is from the achievements made by G. Perelman with this technique in 2002 when the Ricci flow has been established in a discipline itself, generating a great interest in the community. This thesis contains four original results. First result…

Curvature tensor under the Ricci flow

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. We prove that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.

Normalized Ricci flow on nonparabolic surfaces

This paper studies 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.

四维梯度Ricci孤立子的几何与分析

Geometry and Analysis of Gradient Ricci Solitons in Dimension Four

本文综述了四维Gradient Ricci solitons的几何性质、渐近行为及分类与刚性的最新进展。

A lower bound for the diameter of solutions to the Ricci flow with nonzero $H^{1}(M^{n};R)$

We obtain a lower bound for the diameter of a solution to the Ricci flow on a compact manifold with nonvanishing first real cohomology. A consequence of our result is an affirmative answer to Hamilton's conjecture that a product metric on $(S^{1}\times S^{n-1}$ cannot arise as a final time limit flow.

Displacement interpolations from a Hamiltonian point of view

One of the most well-known results in the theory of optimal transportation is the equivalence between the convexity of the entropy functional with respect to the Riemannian Wasserstein metric and the Ricci curvature lower bound of the underlying Riemannian manifold. There are also generalizations of this result to the Finsler manifolds and manifolds with a Ricci flow background. In this paper, we study displacement interpolations from the point of view of Hamiltonian systems and give a unifying approach to the above mentioned results.

Ricci flow and the holonomy group

We prove that the restricted holonomy group of a complete smooth solution to the Ricci flow of uniformly bounded curvature cannot spontaneously contract in finite time; it follows, then, from an earlier result of Hamilton that the holonomy group is exactly preserved by the equation. In particular, a solution to the Ricci flow may be Kähler or locally reducible (as a product) at $t= T$ if and only if the same is true of $g(t)$ at times $t\leq T$.

Foliation divisorial contraction by the Sasaki-Ricci flow on Sasakian 5-manifolds

Let (M,η,ξ,Φ,g) be a compact quasi-regular Sasakian 5-manifold with finite cyclic quotient foliation singularities of type (1/r)(1,a). First, we derive the foliation minimal model program by applying the resolution of cyclic quotient foliation singularities. Secondly, based on the study of local model of resolution of foliation singularities, we prove the foliation canonical surgical contraction or the foliation extremal ray contraction under the Sasaki-Ricci flow. As a consequence, we prove a Sasaki analogue of analytic minimal model program with the Keahler-Ricci flow due to Song-Tian and So…

Ricci Flow of Compact Locally Homogeneous Geometries on 5-Manifolds

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…

On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows

In this paper, we prove the Li-Yau type Harnack inequality and Hamilton type dimension free Harnack inequality for the heat equation $\partial_t u=Lu$ associated with the time dependent Witten Laplacian on complete Riemannian manifolds equipped with a variant of the $(K, m)$-super Perelman Ricci flows and the $K$-super Perelman Ricci flows.