曲面上初始曲率无下界情形Ricci flow的存在性、唯一性
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摘要
1982年,Richard Hamilton在论文[H1]中引入了Ricci flow方程开创了Ricci flow理论的研究。Ricci flow作为一门分析工具,在几何和物理等领域的研究中起了重要应用,如证明三维Poincare猜想、研究天体物理学中的黑洞理论等。同时,其作为一个抛物型偏微分方程,研究在各种条件下解的存在性、唯一性,也是有极大意义的。所以,自从1982年Hamilton在[H1]中证明了闭流形上Ricci flow的短时间存在性、唯一性之后,对Ricci flow解的存在性、唯一性研究直到现在都是数学家们关注的热点。
Hamilton在[H1]中证明了如下定理
定理0.0.1.
如果Mn是一个闭的黎曼流形,且g0是一个光滑的黎曼度量,则存在唯一的Ricci flow g(t),t∈[0,δ),δ>0,且g(0)=g0.
Hamilton克服了方程(1.1)的不严格抛物性得到了该定理,但其证明非常繁琐。后来,经DeTurck改进,用Ricci-DeTurck flow的方法将证明大大简化,此方法后来也成为了研究Ricci flow的标准方法。参见[DT]。
1989年,华人数学家Wanxiong Shi用DeTurck flow的方法,将Hamilton的工作推广到非紧流形,证明了非紧流形上Ricci flow的短时间存在性定理：
定理0.0.2.
([S1])设(Mn,g0)是一个完备非紧的光滑黎曼流形,且黎曼曲率张量Rm的有界,则存在一个完备Ricci flow g(t), t∈[0,T),使得g(0)=g0,且有supM×[0,T)|Rm|=∞或者T=∞。
完备非紧流形上解的唯一性问题,近年来一直是深受数学家们关注的问题。在[LT]中,Peng Lu和Gang Tian用DeTurck技巧,证明了Ricci flow在(?)n,n≥3情况下关于原点旋转对称的Ricci flow的唯一性。后来,Hsu在[Hs]中拓展了Lu-Tian的结果,证明了与标准Ricci flow相关的Ricci harmonic flow的旋转对称解的唯一性。
但更一般性的结果,是2006年陈兵龙和朱熹平在[CZ]中证明的：
定理0.0.3.
([CZ])设(M,g)是一个完备非紧的光滑黎曼流形,Rm(g)有界。假设存在两个Ricci flow g1(t)和g2(t), t∈[0,T],且g1(0)= g2(0)= g。如果Rm(g1(t))和Rm(g2(t))在t∈[0,T]都有界,则g1 (t)＝g2(t)。
由于Ricci flow理论在数学、物理等领域有重要应用,我们有必要研究初始条件进一步弱化时Ricci flow的存在性、唯一性。2007年,美国数学家Peter Topping在[T1]中首次尝试给出曲面上初始曲率无界情形Ricci flow的存在性结果。
定理A设M是一个带有光滑黎曼度量g的二维开曲面,高斯曲率K(g)有上界K。存在仅依赖于K的T>0,使得在M上存在一个光滑Ricci flow g(t),g(0)=g,t∈[0,T],且对任意t0> 0, K(g(t))在[t0,T]上有界。
我们主要研究二维情形初始Gauss曲率仅有上界是Ricci flow的存在性和唯一性问题。由于Topping关于定理A的证明并不完整,我们利用Pseudolocality定理和极大值原理在本文第四章给出一个完备证明。
在此基础上,我们利用Ricci flow在二维时保持初始度量共形类的特点,用线性化的方法证明了解在一定条件假设下的唯一性。(参见[CY],[T1])具体定理如下：
定理B设(M,g(0))是一个二维非紧完备流形,其高斯曲率K(0)仅有上界K,K≤0.如果g1(t)与g2(t)在M×[0,T]上均是Ricci flow,且有相同初始度量g(0),0 以及
定理C设(M,g(0))是一个二维完备非紧流形,高斯曲率K(0)仅有上界K,K>0.如果g1(t)和g2(t)在M×[0,T]都是Ricci flow,有相同初始度量9(0),0 当K>0时,我们还有以下定理：
定理D设(M,g(0))为一个二维完备非紧流形,高斯曲率K(0)仅有上界K,K>0。若g1(t)和g2(t)在M×[0,T]均是Ricci flow,且有相同初始度量g(0),0 3.存在某紧区域∑和一个很小的时间区间[0,t0](?)[0,T]使得Ki(x,t)对(x,t)∈(M\Σ)×[0,t0]非正,i=1,2,那么g1(x,t)=g2(x,t),(?)(x,t)∈M×[0,T]。
这些唯一性定理,我们将在本文第五章给出叙述和证明。
In 1982, Richard Hamilton in introduced Ricci flow equaiton in[H1], and started the research of Ricci flow theory. As an important analyzing tool, Riccci flow theory is widely applied in Geometrical Analysis and Physics, such as the proof of Poincare conjecture、the research of black hole in astrophysics etc. On the other hand, as a parabolic equation, the research of existence and uniqueness of Ricci flow under various of initial conditions is also very important to the development of equation field. For this reason, since Hamilton proved the short-time existence and uniqueness of Ricci flow on closed manifolds in[H1], the research of existence and uniqueness of Ricci flow is still a front topic today.
In[H1] Hamilton proved the following theorem:
Theorem 0.0.4.
If(Mn, g0) is a closed Riemannian manifold, with go smooth, then there exists a unique Ricci flow g(t),t∈[0,δ),δ> 0, with g(0)= g0.
The theorem is very great, though the original proof is very complicated. Not long after, DeTurck improved Hamilton's proof by using the Ricci-DeTurck flow, and this method has become a standard method in Ricci flow research.
In 1989, Wanxiong Shi extended Hamilton work to noncompact case, and proved the following theorem:
Theorem 0.0.5.
([S1])Let (Mn, g0) be a complete noncompact Riemannian manifold, with Riemannian curvature Rm bounded, then there exists a complete Ricci flow g(t), t∈[0, t), such that g(0)= g0,and supM×[0,T)|Rm|=∞or T=∞。
On the research of uniqueness of Ricci flow on complete noncompact manifold, there have been plenty of results these years too. In[LT], Peng Lu and Gang Tian used DeTurck's trick to prove the uniqueness of the standard solution of Ricci flow on Rn, n≥3, which is radially symmetric about the origin. Hsu extended Lu and Tian's result in [Hs], and proved the uniqueness of the solution of the radically symmetric solution of the Ricci harmonic flow associated with the standard solution of Ricci flow.
Nevertheless, the first general uniqueness result is proved by Bing-Long Chen and Xi-Ping Zhu in 2006:
Theorem 0.0.6.
([CZ]) Let (M, g) be a complete noncompact smooth Riemannian manifold, with Rm(g) bounded. Suppose there exist two Ricci flow g1(t) and g2(t), with t∈[0,T], and g1 (0)= g2(0)= g. If both Rm(g1(t)) and Rm(g2(t)) are bounded for t∈[0,T], then g1 (t)＝g2(t).
Due to the importance of Ricci flow in both mathematical and physical field, it's necessary to study the existence and uniqueness of Ricci flow under weaker initial conditions.
In 2007, Peter Topping firstly tried to prove the existence of Ricci flow on surfaces with unbounded initial curvature in [T1]. This is a very interesting try, though there were a few defaults inside. Topping's theorem is as follows:
Theorem A Let M be an open surface equipped with a smooth metric g, and the Gaussian curvature is bounded above only. Then, there exists a constant T> 0 depending only on the supremum of K(g), such that a smooth Ricci flow g(t) exists on M for t∈[0, T], and the Gaussian curvature is bounded (?)t∈[t0, T], (?)t0> 0.
In Chapter 4 of this paper, we give a complete proof, see also in[CY].
Besides, as Ricci flow in dimension 2 remains the conformal class of initial met-ric, we can simplify the Ricci flow equation and prove the uniqueness under certain assumption.
The uniqueness theorems will be proved are:
Theorem B Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) only bounded above by K, K< 0. If both g1(t) and g2(t) are Ricci flow solutions on M×[0, T] with the same initial metric g(0),0< T<∞, and satisfy for some positive constant C, here po(p,x) means the distance between x and a fixed point p with respect to initial metric g(0),i=1,2, then g1(x,t)＝g2(x,t),(?) (x,t)∈M×[0,T].
Besides, we still have
Theorem C Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) only bounded above by K, K> 0. If both g1(t) and g2(t) are Ricci flow solutions on M×[0, T] with the same initial metric g(0),0< T< K-1/2, and satisfy then g1(x,t)=g2(x,t),(?)(x,t)∈M×[0,T].
When K> 0, we have the following theorem:
Theorem D Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) bounded above by K, K> 0. If now g1(t) and g2(t) are both Ricci flow, for (x, t)∈M×[0, T], with g(0),0< T< K/2,and they both satisfy
3. there exists some domain E and a small time interval [0, t0](?) [0, T], such that Ki(x, t) is nonpositive for (x, t)∈(M\Σ)×[0, t0],i=1,2, then g1(x, t)= g2(x,t), (?)(x, t)∈M×[0, T].
All the uniqueness theorems mentioned above will be proved in chapter 5.
Hamilton在[H1]中证明了如下定理
定理0.0.1.
如果Mn是一个闭的黎曼流形,且g0是一个光滑的黎曼度量,则存在唯一的Ricci flow g(t),t∈[0,δ),δ>0,且g(0)=g0.
Hamilton克服了方程(1.1)的不严格抛物性得到了该定理,但其证明非常繁琐。后来,经DeTurck改进,用Ricci-DeTurck flow的方法将证明大大简化,此方法后来也成为了研究Ricci flow的标准方法。参见[DT]。
1989年,华人数学家Wanxiong Shi用DeTurck flow的方法,将Hamilton的工作推广到非紧流形,证明了非紧流形上Ricci flow的短时间存在性定理：
定理0.0.2.
([S1])设(Mn,g0)是一个完备非紧的光滑黎曼流形,且黎曼曲率张量Rm的有界,则存在一个完备Ricci flow g(t), t∈[0,T),使得g(0)=g0,且有supM×[0,T)|Rm|=∞或者T=∞。
完备非紧流形上解的唯一性问题,近年来一直是深受数学家们关注的问题。在[LT]中,Peng Lu和Gang Tian用DeTurck技巧,证明了Ricci flow在(?)n,n≥3情况下关于原点旋转对称的Ricci flow的唯一性。后来,Hsu在[Hs]中拓展了Lu-Tian的结果,证明了与标准Ricci flow相关的Ricci harmonic flow的旋转对称解的唯一性。
但更一般性的结果,是2006年陈兵龙和朱熹平在[CZ]中证明的：
定理0.0.3.
([CZ])设(M,g)是一个完备非紧的光滑黎曼流形,Rm(g)有界。假设存在两个Ricci flow g1(t)和g2(t), t∈[0,T],且g1(0)= g2(0)= g。如果Rm(g1(t))和Rm(g2(t))在t∈[0,T]都有界,则g1 (t)＝g2(t)。
由于Ricci flow理论在数学、物理等领域有重要应用,我们有必要研究初始条件进一步弱化时Ricci flow的存在性、唯一性。2007年,美国数学家Peter Topping在[T1]中首次尝试给出曲面上初始曲率无界情形Ricci flow的存在性结果。
定理A设M是一个带有光滑黎曼度量g的二维开曲面,高斯曲率K(g)有上界K。存在仅依赖于K的T>0,使得在M上存在一个光滑Ricci flow g(t),g(0)=g,t∈[0,T],且对任意t0> 0, K(g(t))在[t0,T]上有界。
我们主要研究二维情形初始Gauss曲率仅有上界是Ricci flow的存在性和唯一性问题。由于Topping关于定理A的证明并不完整,我们利用Pseudolocality定理和极大值原理在本文第四章给出一个完备证明。
在此基础上,我们利用Ricci flow在二维时保持初始度量共形类的特点,用线性化的方法证明了解在一定条件假设下的唯一性。(参见[CY],[T1])具体定理如下：
定理B设(M,g(0))是一个二维非紧完备流形,其高斯曲率K(0)仅有上界K,K≤0.如果g1(t)与g2(t)在M×[0,T]上均是Ricci flow,且有相同初始度量g(0),0
定理C设(M,g(0))是一个二维完备非紧流形,高斯曲率K(0)仅有上界K,K>0.如果g1(t)和g2(t)在M×[0,T]都是Ricci flow,有相同初始度量9(0),0
定理D设(M,g(0))为一个二维完备非紧流形,高斯曲率K(0)仅有上界K,K>0。若g1(t)和g2(t)在M×[0,T]均是Ricci flow,且有相同初始度量g(0),0
这些唯一性定理,我们将在本文第五章给出叙述和证明。
In 1982, Richard Hamilton in introduced Ricci flow equaiton in[H1], and started the research of Ricci flow theory. As an important analyzing tool, Riccci flow theory is widely applied in Geometrical Analysis and Physics, such as the proof of Poincare conjecture、the research of black hole in astrophysics etc. On the other hand, as a parabolic equation, the research of existence and uniqueness of Ricci flow under various of initial conditions is also very important to the development of equation field. For this reason, since Hamilton proved the short-time existence and uniqueness of Ricci flow on closed manifolds in[H1], the research of existence and uniqueness of Ricci flow is still a front topic today.
In[H1] Hamilton proved the following theorem:
Theorem 0.0.4.
If(Mn, g0) is a closed Riemannian manifold, with go smooth, then there exists a unique Ricci flow g(t),t∈[0,δ),δ> 0, with g(0)= g0.
The theorem is very great, though the original proof is very complicated. Not long after, DeTurck improved Hamilton's proof by using the Ricci-DeTurck flow, and this method has become a standard method in Ricci flow research.
In 1989, Wanxiong Shi extended Hamilton work to noncompact case, and proved the following theorem:
Theorem 0.0.5.
([S1])Let (Mn, g0) be a complete noncompact Riemannian manifold, with Riemannian curvature Rm bounded, then there exists a complete Ricci flow g(t), t∈[0, t), such that g(0)= g0,and supM×[0,T)|Rm|=∞or T=∞。
On the research of uniqueness of Ricci flow on complete noncompact manifold, there have been plenty of results these years too. In[LT], Peng Lu and Gang Tian used DeTurck's trick to prove the uniqueness of the standard solution of Ricci flow on Rn, n≥3, which is radially symmetric about the origin. Hsu extended Lu and Tian's result in [Hs], and proved the uniqueness of the solution of the radically symmetric solution of the Ricci harmonic flow associated with the standard solution of Ricci flow.
Nevertheless, the first general uniqueness result is proved by Bing-Long Chen and Xi-Ping Zhu in 2006:
Theorem 0.0.6.
([CZ]) Let (M, g) be a complete noncompact smooth Riemannian manifold, with Rm(g) bounded. Suppose there exist two Ricci flow g1(t) and g2(t), with t∈[0,T], and g1 (0)= g2(0)= g. If both Rm(g1(t)) and Rm(g2(t)) are bounded for t∈[0,T], then g1 (t)＝g2(t).
Due to the importance of Ricci flow in both mathematical and physical field, it's necessary to study the existence and uniqueness of Ricci flow under weaker initial conditions.
In 2007, Peter Topping firstly tried to prove the existence of Ricci flow on surfaces with unbounded initial curvature in [T1]. This is a very interesting try, though there were a few defaults inside. Topping's theorem is as follows:
Theorem A Let M be an open surface equipped with a smooth metric g, and the Gaussian curvature is bounded above only. Then, there exists a constant T> 0 depending only on the supremum of K(g), such that a smooth Ricci flow g(t) exists on M for t∈[0, T], and the Gaussian curvature is bounded (?)t∈[t0, T], (?)t0> 0.
In Chapter 4 of this paper, we give a complete proof, see also in[CY].
Besides, as Ricci flow in dimension 2 remains the conformal class of initial met-ric, we can simplify the Ricci flow equation and prove the uniqueness under certain assumption.
The uniqueness theorems will be proved are:
Theorem B Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) only bounded above by K, K< 0. If both g1(t) and g2(t) are Ricci flow solutions on M×[0, T] with the same initial metric g(0),0< T<∞, and satisfy for some positive constant C, here po(p,x) means the distance between x and a fixed point p with respect to initial metric g(0),i=1,2, then g1(x,t)＝g2(x,t),(?) (x,t)∈M×[0,T].
Besides, we still have
Theorem C Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) only bounded above by K, K> 0. If both g1(t) and g2(t) are Ricci flow solutions on M×[0, T] with the same initial metric g(0),0< T< K-1/2, and satisfy then g1(x,t)=g2(x,t),(?)(x,t)∈M×[0,T].
When K> 0, we have the following theorem:
Theorem D Let (M, g(0)) be a 2-dimensional complete noncompact manifold, with Gaussian curvature K(0) bounded above by K, K> 0. If now g1(t) and g2(t) are both Ricci flow, for (x, t)∈M×[0, T], with g(0),0< T< K/2,and they both satisfy
3. there exists some domain E and a small time interval [0, t0](?) [0, T], such that Ki(x, t) is nonpositive for (x, t)∈(M\Σ)×[0, t0],i=1,2, then g1(x, t)= g2(x,t), (?)(x, t)∈M×[0, T].
All the uniqueness theorems mentioned above will be proved in chapter 5.
引文
[C]E.Calabi, An extension of E.Hopf's maximum principle with an application to geometry, Duke Math J.25(1965),285-303.
[Ch]Isaac Chavel, Eigenvalues in Riemannian geometry, Acadamic press inc.1984.
[Chern]陈省身,陈维恒,微分几何讲义。北京大学出版社,2003年。
[ChenYZ]Yazhe Chen,2-order parabolic type partial differential equation, Peiking university press.
[CH]Albert Chau,Luen-Fai Tam,ChengJie Yu, Pseudolocality for the Ricci flow and applications. arxiv:math/0701153v2.
[CK]Bennett Chow, Dan Knopf, The Ricci flow:An Introduction. American Mathematical Soci-ety.2004.
[CLN]Bennett Chow, Peng Lu, Lei Ni, Hamilton's Ricci flow. American mathematical soci-ety.,(2005).
[CY]Qing Chen, Yajun Yan, Ricci flow on surfaces with unbounded initial metric. Submitted.
[CY2]Qing Chen, Yajun Yan, Uniqueness of Ricci flow with unbounded initial curvature in dimen-sion 2. Accepted by AGAG.
[CZ]Binglong Chen, Xiping Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds J. Differential Geom. Volume 74, Number 1 (2006),119-154.
[DT]D.M.DeTurck,Deforming the metrics in the direction of Ricci tensors(improved version), in collected papers on Ricci flow, International press,2003.
[Do]Do Carmo, Riemannian Geometry. Birkhauser,1993.
[D]J.Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, U niv.Math.J.32(1983)703-716.
[H1]R.Hamilton, Three-manifolds with positive Ricci curvature, J.Differential Geometry 17(1982) 255-306.
[H2]R.S.Hamilton, Ricci deformation of the metric on a Riemannian manifold. J.Differential.Geom.21,(1985),47-62.
[H3]R.H.Hamilton, A compactness property for solutions of the Ricci flow. American Journal of Mathematics.117 (1995)545-572.
[Hs]Shu-Yu Hsu,/abs/math.DG/07021168
[Hsul]Shuyu Hsu, Uniqueness of solutions of Ricci flow on complete noncompact manifolds, arXiv:math/0704.3468v2.
[Hsu2]Shuyu Hsu,Maximum principle and convergence of fundamental solutions for the Ricci flow http://arxiv.org/abs/0711.1236.
[Hsu3]Shu-Yu Hsu, A Pseudolocality theorem for Ricci flow. arxiv:0908.0869v1.
[Jost]Jurgen Jost, Riemannian Geometry and Geometric Analysis. Springer press.2002.
[KZ]S.Kuang,Q.S.Zhang,A gradient estimete for all posieive solutions of the conjugate heat equa-tion under Ricci flow J.Funct.Anal.255(2008),No.4,1008-1023.
[L]Bruce Kleiner, John Lott, Notes on Perelman's papersarxiv.math/0605667v2.
[LT]Peng Lu, Gang Tian,/lott/ricciflow/StanUniq Work2.pdf
[Lu]P Lu,Local curvature bound in Ricci flow. http://arxiv.org/abs/0906.3784
[MS1]Miles Simon,Ricci flow of almost non-negatively curved three manifolds, arXiv: math/0612095.
[MS2]Miles Simon,Ricci flow of non-collapsed 3-manifolds with Ricci curvature is bounded form below, arXiv:math/0903.2142.
[MT]John W.Morgan,Gang Tian, Ricci flow and the Poincare Conjecture.2006
[P]G.Perelman, The entropy formula for the Ricci flow and its geometric application. arxiv:math.DG/0211159v1. (2002)
[PC]彭家贵,陈卿,微分几何讲义。高等教育出版社。2002年7月第一版。
[S1]Wanxiong Shi, Deforming the metric on complete Riemannian mani-folds.J.Differential.Geom.30,(1989), no.1,223-301.
[S2]Wanxiong Shi, Ricci deformation of the metric on complete noncompact Riemannian mani-folds.J.Differential.Geom.30,(1989), no.2,303-394.
[S3]Wanxiong Shi,Ricci flow and the uniformizaitonon complete Kahler mani-foldsJ.Differential.Geom.45(1997).94-220.
[T1]Peper Topping, Ricci flow compactness via psedolociality,and flows with incomplete initial metrics,(2006). To appear.
[Wu]伍鸿熙,沈纯理,虞言林,黎曼几何初步,北京大学出版社。1989年。
[Y]Shing-Tung Yau, Richard Schoen, Lectures on differential geometry, International Press,1994. 5
[Ch]Isaac Chavel, Eigenvalues in Riemannian geometry, Acadamic press inc.1984.
[Chern]陈省身,陈维恒,微分几何讲义。北京大学出版社,2003年。
[ChenYZ]Yazhe Chen,2-order parabolic type partial differential equation, Peiking university press.
[CH]Albert Chau,Luen-Fai Tam,ChengJie Yu, Pseudolocality for the Ricci flow and applications. arxiv:math/0701153v2.
[CK]Bennett Chow, Dan Knopf, The Ricci flow:An Introduction. American Mathematical Soci-ety.2004.
[CLN]Bennett Chow, Peng Lu, Lei Ni, Hamilton's Ricci flow. American mathematical soci-ety.,(2005).
[CY]Qing Chen, Yajun Yan, Ricci flow on surfaces with unbounded initial metric. Submitted.
[CY2]Qing Chen, Yajun Yan, Uniqueness of Ricci flow with unbounded initial curvature in dimen-sion 2. Accepted by AGAG.
[CZ]Binglong Chen, Xiping Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds J. Differential Geom. Volume 74, Number 1 (2006),119-154.
[DT]D.M.DeTurck,Deforming the metrics in the direction of Ricci tensors(improved version), in collected papers on Ricci flow, International press,2003.
[Do]Do Carmo, Riemannian Geometry. Birkhauser,1993.
[D]J.Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, U niv.Math.J.32(1983)703-716.
[H1]R.Hamilton, Three-manifolds with positive Ricci curvature, J.Differential Geometry 17(1982) 255-306.
[H2]R.S.Hamilton, Ricci deformation of the metric on a Riemannian manifold. J.Differential.Geom.21,(1985),47-62.
[H3]R.H.Hamilton, A compactness property for solutions of the Ricci flow. American Journal of Mathematics.117 (1995)545-572.
[Hs]Shu-Yu Hsu,/abs/math.DG/07021168
[Hsul]Shuyu Hsu, Uniqueness of solutions of Ricci flow on complete noncompact manifolds, arXiv:math/0704.3468v2.
[Hsu2]Shuyu Hsu,Maximum principle and convergence of fundamental solutions for the Ricci flow http://arxiv.org/abs/0711.1236.
[Hsu3]Shu-Yu Hsu, A Pseudolocality theorem for Ricci flow. arxiv:0908.0869v1.
[Jost]Jurgen Jost, Riemannian Geometry and Geometric Analysis. Springer press.2002.
[KZ]S.Kuang,Q.S.Zhang,A gradient estimete for all posieive solutions of the conjugate heat equa-tion under Ricci flow J.Funct.Anal.255(2008),No.4,1008-1023.
[L]Bruce Kleiner, John Lott, Notes on Perelman's papersarxiv.math/0605667v2.
[LT]Peng Lu, Gang Tian,/lott/ricciflow/StanUniq Work2.pdf
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