Ricci流极限的性质
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摘要
1982年,Hamilton在他的开创性论文中创立了Ricci流,从此之后,Ricci流就成为学习黎曼几何性质的强大工具。Perelman继续Hamilton的工作,利用Ricci流最终解决了Poincare猜想,请见Perelman的文章[79,80,81].除此之外,利用Ricci流,还给出了黎曼曲面单值化定理新的证明[16,49,23],解决了具有正曲率算子的紧流形的分类[2],和著名的1/4-pinchching定理[1],等等。。。
Perelman的文章[79,80,81]中,有很多断言,但没有证明或没有提供详细证明。例如,[79]里的推论9.3,Perelman只提供了一个证明的重点,后来L.Ni利用Perelman的诱导距离的性质,第一个给出了这个推论的详细证明。最近,Chau,Tam和余在他们的文章[29]中也给出了这个推论的另一个详细证明,他们主要通过利用基本解的估计。另一方面,对于具有非负Ricci曲率的闭流形,L.Ni在2004年已经证明了一个类似的结果。本论文的第三章,按照Chau,Tam和余的方法,我们得到了本论文的第一个主要定理。
定理A.对于某个T>0,假设(M,9(t))是M×[0,T]的一个超级Ricci流,即满足(?)gij=hij≤Rij,(T=T-t),还假设满足以下条件
1.第二Bianchi恒等式,
2.热方程型不等式这里H=gijhij,我们还假设|(?)kRm|和|(?)kh|(k=0,1,2)是有界的。假设p是M上的固定点,Z(p,T；χ，t)是共轭热方程的正基本解,中心为(p,T),即,limt→TZ(p,T；χ,t)=δp°令u(χ,t)=Z(p,T；χ,t),且u=(?),则有
以上定理的证明关键点是对于满足以上条件的超级Ricci流的正基本解具有相同的估计以及我们在以上条件下可以导出一个好的”单调性“公式。
本论文的第四章,我们主要是对Perelman的断言之一提供了详细证明。
定理B.令(Mk,gk(t),χk)是一系列点状Ricci流(?)=-2Rij的光滑解,这里的Mk具有有界曲率并且(Mk,gk(t))在t∈[0,T]上是完备的。假设对于某个K>0和在Cheeger-Gromo意义下光滑收敛到Ricci流的一个光滑解
定义Mk×[0,T)上的共轭热方程,(-(?)-△+Rk)u=0,的极小正基本解uk满足,当时间接近于丁时,收敛到以χk为中心的δ-函数；也就是说uk是(-(?)-△+Rk)u=0上极小正解且那么存在Ф(?)(uk)的一个子列在M∞×(0,T)上的每一个紧子集上一致收敛到M∞×(0,T)上共轭热方程的一个极小正基本解u,这个基本解满足当时间趋近于T时收敛到以p点为中心的δ-函数。
这个定理的证明主要是利用了[29]中正基本解的相关估计。
本论文的第五章,我们给出了在Ricci流下的数量曲率的一个局部下界估计和Ricci solitons的一些性质。首先,对应于陈兵龙的关于满足Ricci流的数量曲率的整体下界估计的结果,我们得到了一个局部的结果,通过利用Yokota在文章[99]中的方法。这是本论文的第三个主要定理。
定理c.对于任意的0<ε<(?)。假设(Mn,g(t)),t∈[α,β]是Ricci流的一个完备解,给定M上的p点,那么存在只依赖于p和度量g(t),t∈[α,β]的常数C(p)和常数C,使得当c≥C(p)时,对于χ∈Bg(t)(p,c),t∈(α,β]就有成立,这里A(ε)=2/n-ε,B(ε)=3C/2√Aεc2。
由这个定理可以导出两个推论。第一个是数量曲率的整体下界估计。
推论C1.假设(Mn,g(t)),t∈[α,β],是Ricci流的一个完备解,则对于任何的t∈(α,β)有
由于ancient solution是Ricci流的一个特殊解,具有t∈(-∞,0]。从而我们有下面的性质。
推论C2.假设(Mn,g(t)),t∈(-∞,0],是Ricci流的一个ancient solution,则对于任何的t∈(-∞,0],有
下面,我们要学习梯度Ricci solitons的一些性质。在很多人对Ricci solitons的学习之后,我们已经知道了很多Ricci solitons的性质。
对于黎曼流形(Mn,g)和Mn上的光滑函数f以及常数ε∈R,如果满足我们则称(Mn,g,f,ε)是梯度Ricci soliton。称f为势函数。如果ε<0,ε=0,或者ε>0,我们称g分别是收缩的,稳定的,或者扩张的。
下面的性质是陈兵龙在[34]中的一个性质的直接结果,但我们利用Ricci soli-tons的方程给出了一个直接的不同于陈兵龙的证明方法。
定理D.假设(Mn,g,f,ε)是一个非紧的完备梯度Ricci soliton。有下列性质
(1)如果梯度Ricci soliton是收缩的,则R≥O。甚至,如果数量曲率在某点等于0,则(Mn,g)等距同构Rn。
(2)如果梯度Ricci soliton是稳定的,则R≥0。甚至,如果数量曲率在某点等于0,则(Mn：g)是Ricci平坦流形。
(3)如果梯度Ricci soliton是扩张的,则R≥-(?)。甚至,如果数量曲率在某点等于-(?),则(Mn,g)是Einstein流形。
对于收缩Ricci soliton具有最大欧式体积增长,已经分别被曹怀东和Zhou [32], Munteanu [66]得到。如果假设数量曲率有一致下界,我们将得到收缩Ricci soli-ton具有最多rσ(σ 定理E.假设(Mn,g,f,-1)是具有R≥δ>0的收缩梯度Ricci soliton。那么给定ο∈Mn,存在一个只依赖于δ,ο和Ricci soliton本身的常数C<∞,使得对所有的r≥0,都有
由这个定理很容易推出Carrillo和L.Ni[26]中的一个结果。
推论E1.任意具有非负Ricci曲率的非平坦的收缩梯度Ricci soliton一定有V(g)=0。
注意V(g)是在具有非负Ricci曲率的流形上定义为
现在,3维收缩梯度Ricci solitons已经完全被分类了。本论文的最后一个主要定理是证明了Petersen和Wylie[83]中的某个定理的一个条件在收缩Ricci soli-ton下自然成立。
定理F.假设(M,g,f,-1)是完备收缩梯度Ricci soliton,则有甚至,有
推论F1.假设(M,g,f,-1)是具有W=0的n(n≥3)维完备收缩梯度Ricci soliton，则M一定是Sn,Rn或Sn-1×R的一个有限商空间。特别地,3维收缩梯度Ricci soliton一定是S3,R3或S2×R的一个有限商空间。
In 1982, Ricci flow was introduced in Hamilton's seminal paper [47]. Since then, Ricci flow has become a powerful tool to study Riemannian geometry. Fol-lowing Hamilton's program, G. Perelman recently solved the Poincare conjecture, which proposes a topological characterization of the 3-sphere, and Thurston's ge-ometrization conjecture [79,80,81]. Besides this, Ricci flow has been used to solve several other important problems, such as a new proof of the uniformization of Riemannian surfaces [16,49,23], the classification of manifolds with positive curvature operator [2], and the 1/4-pinching theorem [1], etc.
In Perelman's paper [79], there are many claims without detailed proofs. For example, Corollary 9.3, Ni first gave its detailed proof in his paper [77] using the properties of Perelman's reduced distance. Recently, Chau, Tarn and Yu [29] also gave another detailed proof using estimates of fundamental solutions. On the other hand, Ni has proved a similar result for closed manifolds with nonnegative Ricci curvature in 2004. I would like to unify the two inequalities. In the chapter 3 of this thesis, following Chau, Tam and Yu's argument, we obtain our first main theorem.
Theorem A. Let (M, g(t)) be a solution of the super Ricci flow, i.e.,(?)= hij≤Rij,τ= T-t, on M x [0. T] for some T>0, satisfying assumptions where H=gijhij, and bounded |▽kRm| and |▽kh| for k=0,1,2. Let p∈M be a fixed point, and let Z(p,T;x,t) be a positive fundamental solution of the conjugate heat equation centered at (p,T), i.e.,limt→TZ(P,T;x,t)=δp. Let u(x,t)= Z(p,T;x,t) and
The key points of this proof are the same estimates of fundamental solutions under super Ricci flow with above assumptions and a nice "monotony formula"
In the chapter 4 of this thesis, we state the second main theorem, which gives a result about the convergence of fundamental solutions in the Cheeger-Gromov sense under Ricci flow, which was claimed by Perelman [79]. It plays an important role in the proof of Perelman's pseudolocality theorem.
Theorem B. Let (Mk,gk(t),xk) be a sequence of pointed Ricci flow (?)gij-2Rij, where each Mk is a non-compact manifold with bounded curvature such that (Mk,gk(t)) is complete for t∈[0,T]. Suppose for some constant K>0 and converges in the C∞pointed Cheeger-Gromov sense to a smooth solution of Ricci flow,
Define the function uk on Mk×[0,T) to be the minimal positive fundamental solution of the conjugate heat equation, (-(?)-△+Rk)u=0, limiting to theδ-function centered at xk as time approaches T; i.e., uk is the minimal positive solution to (-(?)-△+Rk)u=0 and
ThenΦk*(uk) subconverges uniformly on every compact subset of M∞x (0,T) to the minimal positive fundamental solution u of the conjugate heat equation on M∞×(0,T) limiting to theδ-function centered at p as time approaches T.
The key point is to use some estimates of fundamental solutions proved in [29].
In the chapter 5 of this thesis,we obtain several theorems.Comparing Chen's global lower bound of the scalar curvature for Ricci flow,we obtain a local lower bound of the scalar curvature for Ricci flow,using Yokota's argument,[99].It is our third main theorem.
Theorem C.For any 0<ε<2/n.Suppose(Mn,g(t)), t∈[α,β] is a complete solution to Ricci flow,p∈M,then there exist constants C(p)depending on p and the metrics g(t),t∈[α,β] and C such that when c≥C(p),we have whenever x∈Bg(t)(p,c),t∈(a,β],where A(ε)=2/n-ε, B(ε)=3C/2√Aεc2 .
It implies two corollaries.
Corollary C1.Suppose(Mn,g(t)), t∈[a,β],is a complete solution to the Ricci flow,then on M×(α,β]
Since ancient solution is a special solution to the Ricci flow for t∈(-∞,0]. Then we have the following property.
Corollary C2.If(Mn,g(t)), t∈(-∞,0],is a complete ancient solution to the Ricci flow,then on M×(-∞,O].
From now on,we study some properties of gradient Ricci solitions. The properties of Ricci solitons have been studied by many people.
We say that a quadruple(Mn,g,f,ε),where(Mn,g)is a Riemannian main-fold,f is a smooth function on Mn andε∈R,is a gradient Ricci soliton if We call f the potential function.We say that g is shrinking,steady,or expanding ifε<0,ε=0,orε>0,respectively.
The following property is a consequence of Chen's results in [34], but I will give the direct proof using Ricci solitons equation.
Theorem D. Suppose (Mn. g, f,ε) be a noncompact complete gradient Ricci soliton. Then
(1) If the gradient soliton is shrinking, then R≥0. Moreover, if the scalar curvature obtains 0 at some point, then (Mn,g) is isometric to Rn.
(2) If the gradient soliton is steady, then R≥0. Moreover, if the scalar curvature obtains 0 at some point, then (Mn,g) is Ricci flat.
(3) If the gradient soliton is expanding, then R≥-nε/2. Moreover, if the scalar curvature obtains -nε/2 at some point, then (Mn,g) is Einstein.
The shrinkers have at most Euclidean volume growth was obtained by Cao and Zhou [32] and Munteanu [66]. Add some assumptions about the scalar cur-vature, we also obtain the shrinkers have at most rσ(σ< n) volume growth as our fourth main theorem.
Theorem E. Let (Mn,g,f,-1) be a complete shrinking gradient Ricci soliton with R≥δ>0. Then given o∈Mn, there exists a constant C<∞depending only onδ, o and the soliton such that for all r≥0.
Then we obtain the result which was proved by Carrillo and Ni [26].
Corollary E1. Any nonflat shrinking gradient Ricci soliton with nonnega-tive Ricci curvature must have V(g)=0.
Note that V(g) is defined on manifold with nonnegative Ricci curvature as
The classification of three dimensional shrinking gradient Ricci solitons has been solved after many people's work. Our the last main theorem is the only condition in a result was proved by Petersen and Wylie [83] with vanishing Weyl tensor.
Theorem F. Let (M,g,f,-1) be a complete shrinking gradient Ricci soli-ton, then Moreover,
Corollary F1. Let (M,g,f,-1) be a complete shrinking gradient Ricci soliton of dimension n≥3 with W=0, then M is a finite quotient of Sn,Rn or Sn-1×R. In particular, a three dimensional shrinking gradient Ricci soliton is a finite quotient of S3, R3 or S2×R.
Perelman的文章[79,80,81]中,有很多断言,但没有证明或没有提供详细证明。例如,[79]里的推论9.3,Perelman只提供了一个证明的重点,后来L.Ni利用Perelman的诱导距离的性质,第一个给出了这个推论的详细证明。最近,Chau,Tam和余在他们的文章[29]中也给出了这个推论的另一个详细证明,他们主要通过利用基本解的估计。另一方面,对于具有非负Ricci曲率的闭流形,L.Ni在2004年已经证明了一个类似的结果。本论文的第三章,按照Chau,Tam和余的方法,我们得到了本论文的第一个主要定理。
定理A.对于某个T>0,假设(M,9(t))是M×[0,T]的一个超级Ricci流,即满足(?)gij=hij≤Rij,(T=T-t),还假设满足以下条件
1.第二Bianchi恒等式,
2.热方程型不等式这里H=gijhij,我们还假设|(?)kRm|和|(?)kh|(k=0,1,2)是有界的。假设p是M上的固定点,Z(p,T；χ，t)是共轭热方程的正基本解,中心为(p,T),即,limt→TZ(p,T；χ,t)=δp°令u(χ,t)=Z(p,T；χ,t),且u=(?),则有
以上定理的证明关键点是对于满足以上条件的超级Ricci流的正基本解具有相同的估计以及我们在以上条件下可以导出一个好的”单调性“公式。
本论文的第四章,我们主要是对Perelman的断言之一提供了详细证明。
定理B.令(Mk,gk(t),χk)是一系列点状Ricci流(?)=-2Rij的光滑解,这里的Mk具有有界曲率并且(Mk,gk(t))在t∈[0,T]上是完备的。假设对于某个K>0和在Cheeger-Gromo意义下光滑收敛到Ricci流的一个光滑解
定义Mk×[0,T)上的共轭热方程,(-(?)-△+Rk)u=0,的极小正基本解uk满足,当时间接近于丁时,收敛到以χk为中心的δ-函数；也就是说uk是(-(?)-△+Rk)u=0上极小正解且那么存在Ф(?)(uk)的一个子列在M∞×(0,T)上的每一个紧子集上一致收敛到M∞×(0,T)上共轭热方程的一个极小正基本解u,这个基本解满足当时间趋近于T时收敛到以p点为中心的δ-函数。
这个定理的证明主要是利用了[29]中正基本解的相关估计。
本论文的第五章,我们给出了在Ricci流下的数量曲率的一个局部下界估计和Ricci solitons的一些性质。首先,对应于陈兵龙的关于满足Ricci流的数量曲率的整体下界估计的结果,我们得到了一个局部的结果,通过利用Yokota在文章[99]中的方法。这是本论文的第三个主要定理。
定理c.对于任意的0<ε<(?)。假设(Mn,g(t)),t∈[α,β]是Ricci流的一个完备解,给定M上的p点,那么存在只依赖于p和度量g(t),t∈[α,β]的常数C(p)和常数C,使得当c≥C(p)时,对于χ∈Bg(t)(p,c),t∈(α,β]就有成立,这里A(ε)=2/n-ε,B(ε)=3C/2√Aεc2。
由这个定理可以导出两个推论。第一个是数量曲率的整体下界估计。
推论C1.假设(Mn,g(t)),t∈[α,β],是Ricci流的一个完备解,则对于任何的t∈(α,β)有
由于ancient solution是Ricci流的一个特殊解,具有t∈(-∞,0]。从而我们有下面的性质。
推论C2.假设(Mn,g(t)),t∈(-∞,0],是Ricci流的一个ancient solution,则对于任何的t∈(-∞,0],有
下面,我们要学习梯度Ricci solitons的一些性质。在很多人对Ricci solitons的学习之后,我们已经知道了很多Ricci solitons的性质。
对于黎曼流形(Mn,g)和Mn上的光滑函数f以及常数ε∈R,如果满足我们则称(Mn,g,f,ε)是梯度Ricci soliton。称f为势函数。如果ε<0,ε=0,或者ε>0,我们称g分别是收缩的,稳定的,或者扩张的。
下面的性质是陈兵龙在[34]中的一个性质的直接结果,但我们利用Ricci soli-tons的方程给出了一个直接的不同于陈兵龙的证明方法。
定理D.假设(Mn,g,f,ε)是一个非紧的完备梯度Ricci soliton。有下列性质
(1)如果梯度Ricci soliton是收缩的,则R≥O。甚至,如果数量曲率在某点等于0,则(Mn,g)等距同构Rn。
(2)如果梯度Ricci soliton是稳定的,则R≥0。甚至,如果数量曲率在某点等于0,则(Mn：g)是Ricci平坦流形。
(3)如果梯度Ricci soliton是扩张的,则R≥-(?)。甚至,如果数量曲率在某点等于-(?),则(Mn,g)是Einstein流形。
对于收缩Ricci soliton具有最大欧式体积增长,已经分别被曹怀东和Zhou [32], Munteanu [66]得到。如果假设数量曲率有一致下界,我们将得到收缩Ricci soli-ton具有最多rσ(σ
由这个定理很容易推出Carrillo和L.Ni[26]中的一个结果。
推论E1.任意具有非负Ricci曲率的非平坦的收缩梯度Ricci soliton一定有V(g)=0。
注意V(g)是在具有非负Ricci曲率的流形上定义为
现在,3维收缩梯度Ricci solitons已经完全被分类了。本论文的最后一个主要定理是证明了Petersen和Wylie[83]中的某个定理的一个条件在收缩Ricci soli-ton下自然成立。
定理F.假设(M,g,f,-1)是完备收缩梯度Ricci soliton,则有甚至,有
推论F1.假设(M,g,f,-1)是具有W=0的n(n≥3)维完备收缩梯度Ricci soliton，则M一定是Sn,Rn或Sn-1×R的一个有限商空间。特别地,3维收缩梯度Ricci soliton一定是S3,R3或S2×R的一个有限商空间。
In 1982, Ricci flow was introduced in Hamilton's seminal paper [47]. Since then, Ricci flow has become a powerful tool to study Riemannian geometry. Fol-lowing Hamilton's program, G. Perelman recently solved the Poincare conjecture, which proposes a topological characterization of the 3-sphere, and Thurston's ge-ometrization conjecture [79,80,81]. Besides this, Ricci flow has been used to solve several other important problems, such as a new proof of the uniformization of Riemannian surfaces [16,49,23], the classification of manifolds with positive curvature operator [2], and the 1/4-pinching theorem [1], etc.
In Perelman's paper [79], there are many claims without detailed proofs. For example, Corollary 9.3, Ni first gave its detailed proof in his paper [77] using the properties of Perelman's reduced distance. Recently, Chau, Tarn and Yu [29] also gave another detailed proof using estimates of fundamental solutions. On the other hand, Ni has proved a similar result for closed manifolds with nonnegative Ricci curvature in 2004. I would like to unify the two inequalities. In the chapter 3 of this thesis, following Chau, Tam and Yu's argument, we obtain our first main theorem.
Theorem A. Let (M, g(t)) be a solution of the super Ricci flow, i.e.,(?)= hij≤Rij,τ= T-t, on M x [0. T] for some T>0, satisfying assumptions where H=gijhij, and bounded |▽kRm| and |▽kh| for k=0,1,2. Let p∈M be a fixed point, and let Z(p,T;x,t) be a positive fundamental solution of the conjugate heat equation centered at (p,T), i.e.,limt→TZ(P,T;x,t)=δp. Let u(x,t)= Z(p,T;x,t) and
The key points of this proof are the same estimates of fundamental solutions under super Ricci flow with above assumptions and a nice "monotony formula"
In the chapter 4 of this thesis, we state the second main theorem, which gives a result about the convergence of fundamental solutions in the Cheeger-Gromov sense under Ricci flow, which was claimed by Perelman [79]. It plays an important role in the proof of Perelman's pseudolocality theorem.
Theorem B. Let (Mk,gk(t),xk) be a sequence of pointed Ricci flow (?)gij-2Rij, where each Mk is a non-compact manifold with bounded curvature such that (Mk,gk(t)) is complete for t∈[0,T]. Suppose for some constant K>0 and converges in the C∞pointed Cheeger-Gromov sense to a smooth solution of Ricci flow,
Define the function uk on Mk×[0,T) to be the minimal positive fundamental solution of the conjugate heat equation, (-(?)-△+Rk)u=0, limiting to theδ-function centered at xk as time approaches T; i.e., uk is the minimal positive solution to (-(?)-△+Rk)u=0 and
ThenΦk*(uk) subconverges uniformly on every compact subset of M∞x (0,T) to the minimal positive fundamental solution u of the conjugate heat equation on M∞×(0,T) limiting to theδ-function centered at p as time approaches T.
The key point is to use some estimates of fundamental solutions proved in [29].
In the chapter 5 of this thesis,we obtain several theorems.Comparing Chen's global lower bound of the scalar curvature for Ricci flow,we obtain a local lower bound of the scalar curvature for Ricci flow,using Yokota's argument,[99].It is our third main theorem.
Theorem C.For any 0<ε<2/n.Suppose(Mn,g(t)), t∈[α,β] is a complete solution to Ricci flow,p∈M,then there exist constants C(p)depending on p and the metrics g(t),t∈[α,β] and C such that when c≥C(p),we have whenever x∈Bg(t)(p,c),t∈(a,β],where A(ε)=2/n-ε, B(ε)=3C/2√Aεc2 .
It implies two corollaries.
Corollary C1.Suppose(Mn,g(t)), t∈[a,β],is a complete solution to the Ricci flow,then on M×(α,β]
Since ancient solution is a special solution to the Ricci flow for t∈(-∞,0]. Then we have the following property.
Corollary C2.If(Mn,g(t)), t∈(-∞,0],is a complete ancient solution to the Ricci flow,then on M×(-∞,O].
From now on,we study some properties of gradient Ricci solitions. The properties of Ricci solitons have been studied by many people.
We say that a quadruple(Mn,g,f,ε),where(Mn,g)is a Riemannian main-fold,f is a smooth function on Mn andε∈R,is a gradient Ricci soliton if We call f the potential function.We say that g is shrinking,steady,or expanding ifε<0,ε=0,orε>0,respectively.
The following property is a consequence of Chen's results in [34], but I will give the direct proof using Ricci solitons equation.
Theorem D. Suppose (Mn. g, f,ε) be a noncompact complete gradient Ricci soliton. Then
(1) If the gradient soliton is shrinking, then R≥0. Moreover, if the scalar curvature obtains 0 at some point, then (Mn,g) is isometric to Rn.
(2) If the gradient soliton is steady, then R≥0. Moreover, if the scalar curvature obtains 0 at some point, then (Mn,g) is Ricci flat.
(3) If the gradient soliton is expanding, then R≥-nε/2. Moreover, if the scalar curvature obtains -nε/2 at some point, then (Mn,g) is Einstein.
The shrinkers have at most Euclidean volume growth was obtained by Cao and Zhou [32] and Munteanu [66]. Add some assumptions about the scalar cur-vature, we also obtain the shrinkers have at most rσ(σ< n) volume growth as our fourth main theorem.
Theorem E. Let (Mn,g,f,-1) be a complete shrinking gradient Ricci soliton with R≥δ>0. Then given o∈Mn, there exists a constant C<∞depending only onδ, o and the soliton such that for all r≥0.
Then we obtain the result which was proved by Carrillo and Ni [26].
Corollary E1. Any nonflat shrinking gradient Ricci soliton with nonnega-tive Ricci curvature must have V(g)=0.
Note that V(g) is defined on manifold with nonnegative Ricci curvature as
The classification of three dimensional shrinking gradient Ricci solitons has been solved after many people's work. Our the last main theorem is the only condition in a result was proved by Petersen and Wylie [83] with vanishing Weyl tensor.
Theorem F. Let (M,g,f,-1) be a complete shrinking gradient Ricci soli-ton, then Moreover,
Corollary F1. Let (M,g,f,-1) be a complete shrinking gradient Ricci soliton of dimension n≥3 with W=0, then M is a finite quotient of Sn,Rn or Sn-1×R. In particular, a three dimensional shrinking gradient Ricci soliton is a finite quotient of S3, R3 or S2×R.
引文
[1]S. Brendle and R. Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc.,22 (2009),287-307.
[2]C. Bohm and B. Wilking, Manifolds with positive curvature operators are space forms, Ann. Math.(2),167 (2008),1079-1097.
[3]B. Chow, Expository notes on gradient Ricci solitons, unpublished.
[4]E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke math. J.25 (1965) 45-56.
[5]H.D. Cao, Deformation of Kahler metrics to Kahler-Einstein metrics on compact Kahler manifolds, Inven. Math.,81 (1985),359-372.
[6]H.D. Cao, On Harnack's inequalities for the Kahler-Ricci flow, Inven Math.,109 (1992),247-263.
[7]H.D Cao, Existence of gradient Kahler-Ricci soliton s, Elliptic and Parabolic Meth-ods in Geometry., Minnesota, (1994),1-16.
[8]H.D. Cao, Limits of solutions to the Kahler-Ricci flow, Inven. Math.,45 (1997), 257-272.
[9]H.D. Cao, Geomtery of Ricci solitons, Chinese Annal. Math., Series B,27 (2006), 121-142.
[10]H.D. Cao and B. Chow, Recent developments on the Ricci flow,Bull. AMS.31 (1999),59-74.
[11]J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland publishing Co., Amsterdam,1975.
[12]J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geom.,6 (1972),119-128.
[13]J. Cheeger and M. Gromov, Collapsing Riemannian manifolds-while keeping their curvature bounded, I, J. Diff. Geom.,23 (1986),309-346.
[14]J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded., Ⅱ, J. Diff. Geom.,32 (1990),269-298.
[15]J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom.,17 (1982),15-53.
[16]B. Chow, The Ricci flow on the 2-phere, Jour. Diff. Geom.,33 (1991),325-334.
[17]B. Chow, A survey of Hamilton's program for the Ricci flow on 3-manifolds, Con-temp. Math.,367, Amer. Math. Soc., Providence, RI,2005.
[18]B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, The Ricci Flow:Techniques and Applications, Part Ⅰ: Geometric Aspects. Mathematical Surveys and Monographs, AMS, Providence, RI, 2007.
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[29]A.Chau, L.F.Tam and C.Yu, Pseudolocality for the Ricci flow and applications, arxiv:math.DG/0701153.
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[36]B.L. Chen and X.P. Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, Jour. Diff. Geom.,74 (2006),119-154.
[37]A. Derdzinski, A Myers-type theorem and compact Ricci solitons, Proc. AMS.,134 (2006),3645-3648.
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[40]F.Q. Fang, J.W. Man and Z.L. Zhang, Complete gradient shrinking Ricci solitons have finite topological type, C.R.Acad.Sci.Paris,346 (2008),653-656.
[41]M. Fernandez-Lopez and E. Garcia-Rio, A remark on compact Ricci solitons, Math. Ann.,340 (2007),893-896.
[42]C.M.Guenther, The fundamental solution on manifolds with time-dependent met-rics, J.Geometric Analysis,12(3)(2002),425-436.
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[52]R. Hamilton, A compactness property for solutions of the Ricci flow, Amer. Jour. Math.,117 (1995),545-572.
[53]R. Hamilton, The formation of singulrities in the Ricci flow, Surv. Diff. Geom.,2 (1995),7-163.
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[55]R. Hamilton, Four-manifolds with positive isotropic curvature, Comm Anal. Geom.,5 (1997),1-92.
[56]R. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom.,7 (1999),695-729.
[57]M. Hirsch, Differential Topology, Springer-Verlag (GTM33),1976.
[58]G. Huisken, Ricci deformation of the metric on a Riemannian manifold, Jour. Diff. Geom.,21 (1985),47-62.
[59]S.Hsu, Maximum principle and convergence of fundamental solutions for the Ricci Flow, arxiv:math.DG/0711.1236.
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[61]T. Ivey, Ricci solitons on compact 3-manifolds, Diff. Geom. Appl.,3 (1993),1-11.
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[4]E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke math. J.25 (1965) 45-56.
[5]H.D. Cao, Deformation of Kahler metrics to Kahler-Einstein metrics on compact Kahler manifolds, Inven. Math.,81 (1985),359-372.
[6]H.D. Cao, On Harnack's inequalities for the Kahler-Ricci flow, Inven Math.,109 (1992),247-263.
[7]H.D Cao, Existence of gradient Kahler-Ricci soliton s, Elliptic and Parabolic Meth-ods in Geometry., Minnesota, (1994),1-16.
[8]H.D. Cao, Limits of solutions to the Kahler-Ricci flow, Inven. Math.,45 (1997), 257-272.
[9]H.D. Cao, Geomtery of Ricci solitons, Chinese Annal. Math., Series B,27 (2006), 121-142.
[10]H.D. Cao and B. Chow, Recent developments on the Ricci flow,Bull. AMS.31 (1999),59-74.
[11]J. Cheeger and D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland publishing Co., Amsterdam,1975.
[12]J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geom.,6 (1972),119-128.
[13]J. Cheeger and M. Gromov, Collapsing Riemannian manifolds-while keeping their curvature bounded, I, J. Diff. Geom.,23 (1986),309-346.
[14]J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded., Ⅱ, J. Diff. Geom.,32 (1990),269-298.
[15]J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom.,17 (1982),15-53.
[16]B. Chow, The Ricci flow on the 2-phere, Jour. Diff. Geom.,33 (1991),325-334.
[17]B. Chow, A survey of Hamilton's program for the Ricci flow on 3-manifolds, Con-temp. Math.,367, Amer. Math. Soc., Providence, RI,2005.
[18]B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, The Ricci Flow:Techniques and Applications, Part Ⅰ: Geometric Aspects. Mathematical Surveys and Monographs, AMS, Providence, RI, 2007.
[19]B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, The Ricci Flow:Techniques and Applications, Part Ⅱ: Analytic Aspects. Mathematical Surveys and Monographs, AMS, Providence, RI, 2008.
[20]B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo and L. Ni, The Ricci Flow:Techniques and Applications, Part Ⅲ: Geometric-Analytic Aspects. Mathematical Surveys and Monographs, AMS, Provi-dence, RI,2010.
[21]B. Chow and D. Knopf, The Ricci flow:an introduction, Mathematical surveys and monographs, AMS.,2004.
[22]B.Chow,P.Lu and L.Ni, Hamilton's Ricci flow, Amer. Math. Soc, Providence, RI, 2006.
[23]X.X. Chen, P. Lu and G. Tian, A note on uniformization of Riemann surfaces by Ricci flow, Proc. AMS.,134 (2006),3391-3393.
[24]T. H. Colding and W. P. Minicozzi II, Estimates for the extinction time for the Ricci flow on certain 3-manifolds and a question of Perelman, J. Amer. Math. Soc. 18 (2005),561-569.
[25]T. H. Colding and W. P. Minicozzi II, Width and finite extinction time of Ricci flow,Geom. Topol.12 (2008), no.'5,2537-2586.
[26]J.A. Carrillo and L. Ni, Sharp logarithmic sobolev inequalities on gradient solitons and applications, Comm. Anal. Geom.,17(2009),721-753.
[27]C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ecole Norm. Sup.,13 (1980),419-435.
[28]H. D. Cao and N. Sesum, The compactness result for Kahler Ricci solitons,.Adv. Math.211 (2007), no.2,794-818.
[29]A.Chau, L.F.Tam and C.Yu, Pseudolocality for the Ricci flow and applications, arxiv:math.DG/0701153.
[30]X. Cao, B. Wang and Z. Zhang, On locally conformally flat gradient shrinking Ricci solitons, arXiv:math.DG/0807.0588.
[31]H.D. Cao and X.P. Zhu, A complete proof of the Poincare and geometrization conjectures-application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math.10 (2006) 165-492.
[32]H.D. Cao and D. Zhou, On complete gradient shrinking.solitons, arXiv:math.DG/0903.3932.
[33]H. Chen, Pointwise 1/4-pinched 4-manifolds, Ann. Global Anal. Geom.,9 (1991), 161-176.
[34]B.L. Chen, Strong uniqueness of the Ricci flow, Jour. Diff. Geom.,82 (2009),363-382.
[35]B.L. Chen and X.P. Zhu, Completer Riemannian manifolds with pointwise pinched curvature, Inven. Math.,140 (2000),423-452.
[36]B.L. Chen and X.P. Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, Jour. Diff. Geom.,74 (2006),119-154.
[37]A. Derdzinski, A Myers-type theorem and compact Ricci solitons, Proc. AMS.,134 (2006),3645-3648.
[38]D. DeTurck, Deforming metrics in the direction of their Ricci tensors, Jour. Diff. Geom.,18 (1983),157-162.
[39]M. Feldman, T. Ilmanen and D. Knopf, Rotationally symmetric shrinking and expanding gradient Kahler-Ricci solitons, J. Diff. Geom.,65 (2003),169-209.
[40]F.Q. Fang, J.W. Man and Z.L. Zhang, Complete gradient shrinking Ricci solitons have finite topological type, C.R.Acad.Sci.Paris,346 (2008),653-656.
[41]M. Fernandez-Lopez and E. Garcia-Rio, A remark on compact Ricci solitons, Math. Ann.,340 (2007),893-896.
[42]C.M.Guenther, The fundamental solution on manifolds with time-dependent met-rics, J.Geometric Analysis,12(3)(2002),425-436.
[43]D. Gromoll and W. T. Meyer, Examples of complete manifolds with positive Ricci curvature, J. Differential Geom.21 (1985), no.2,195-211.
[44]M. Gromov, Structures metriques pour les varietes riemanniennes, CEDIC, Paris, 1981, Edit. J. Lafontaine and P. Pansu.
[45]D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, New York (1983).
[46]R.. E. Greene and H. Wu, Lipschitz convergence of Riamannian manifolds, Pacific J. Math.,131 (1988),119-141.
[47]R. Hamilton, Three-manifolds with positive Ricci curvature, Jour. Diff. Geom.,17 (1982),255-306.
[48]R.. Hamilton, Four-manifolds with positive curvature operator, Jour. Diff. Geom. 24 (1986),153-179.
[49]R. Hamilton, The Ricci flow on surfaces, Contem. Math.,71 (1988),237-261.
[50]R. Hamilton, The harnack estimate for the Ricci flow, Jour. Diff. Geom.,37 (1993), 225-243.
[51]R. Hamilton, Eternal solutions to the Ricci flow, J. Diff. Geom.,38 (1993),1-11.
[52]R. Hamilton, A compactness property for solutions of the Ricci flow, Amer. Jour. Math.,117 (1995),545-572.
[53]R. Hamilton, The formation of singulrities in the Ricci flow, Surv. Diff. Geom.,2 (1995),7-163.
[54]R.Hamilton, An isoperimetric estimate for the Ricci flow on the two-sphere,Mod-ern Methods in Complex Analysis, p191-200. Annal. Math. Studies,137,1996.
[55]R. Hamilton, Four-manifolds with positive isotropic curvature, Comm Anal. Geom.,5 (1997),1-92.
[56]R. Hamilton, Non-singular solutions of the Ricci flow on three-manifolds, Comm. Anal. Geom.,7 (1999),695-729.
[57]M. Hirsch, Differential Topology, Springer-Verlag (GTM33),1976.
[58]G. Huisken, Ricci deformation of the metric on a Riemannian manifold, Jour. Diff. Geom.,21 (1985),47-62.
[59]S.Hsu, Maximum principle and convergence of fundamental solutions for the Ricci Flow, arxiv:math.DG/0711.1236.
[60]B.Kleiner and J.Lott, Notes on Perelman's papers, Geom.Topol.12(2008), no.5, 2587-2855.
[61]T. Ivey, Ricci solitons on compact 3-manifolds, Diff. Geom. Appl.,3 (1993),1-11.
[62]T. Ivey, The Ricci flow on radially symmetric R3, Comm. Partial Diff. Equat.19 (1994),1481-1500.
[63]T. Ivey, New examples of complete Ricci solitons, Proc. AMS.,122 (1994),241-245.
[64]P. Li and R. Schoen, LP and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math.153(1984). no3-4,279-301.
[65]P. Li and S.T.Yau, On the parabolic kernel of the Schrodinger operater, Acta Math.156(1986),no.3-4,153-201.
[66]O. Munteanu, The volume growth of complete gradient Shrinking Ricci solitons, arxiv:math.DG/0904.0798.
[67]O. Munteanu and N. Sesum, On gradient Ricci solitons, arxiv:math.DG/0910.1105vl.
[68]J. Milnor, Towards the Poincare conjecture and the classification of 3-manifolds, Notices AMS.,50 (2003),1226-1233.
[69]J. Milnor, Morse Theory, Annals of Mathematics Studies, Princeton University Press,1969.
[70]J. W. Morgan, Recent progress on the Poincare conjecture and the classification of 3-manifolds, Bull. AMS.,42 (2005),57-78.
[71]F. Morgan, Manifolds with density, Notices of AMS.,52 (2005),853-858.
[72]R. McCann and P. Topping, Ricci flow, entropy and optimal transportation, Preprint (2007).
[73]J. W. Morgan and G. Tian, Ricci Flow and the Poincare Conjecture, American Mathematical Society, Clay Mathematics Institute,2007.
[74]A. Naber, Noncompact shrinking 4-solitons with nonnegative curvature, arXiv:0710.5579. [math.DG]
[75]L. Ni, Ancient solutions the Kahler-Ricci flow, Math. Res. Lett.,12 (2005),633-654.
[76]L. Ni, The entropy formula for linear heat equation, J.Geom.Anal., 14 (2004), no. 1,87-100.
[77]L. Ni, A note on Perelman's LYH inequality, Comm. Anal. Geom.14 (2006),883-905.
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