几何流与拓扑的若干问题
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摘要
本文主要有两大部分,第二章和第三章属于几何流部分,剩下的两章为拓扑部分。
近期,几何分析中最重要的发展来自于对几何流方程的研究。其中,最重要的成果是：Huisken与Ilmanen用逆平均曲率流解决了黎曼Penrose不等式和曹怀东与朱熹平用Ricci流工具证明了庞卡莱猜想。
在第二章,我们主要研究了Hk流下拉普拉斯算子的第一特征值,我们首先得到了这个流下拉普拉斯算子第一特征值的发展方程。
我们考虑Rn+1中的紧致无边的,凸的n维超曲面M。令F0:M→Rn+1是到欧式空间的光滑浸入,这里n≥2且H(F0(Mn))>0.M0=F0(M)在Hk流下的演化是一个光滑浸入的单参数族F：M×[0,T]→Rn+1,它满足这里k>0,H是平均曲率且v是外法线方向。我们称上述系统为非规范化的Hk流。注意到k=1,这种流就是非规范后的平均曲率流。
定理A：设Mn(t)为非规范化Hk流的解,令λ=λ(t)为Mn(t)上拉普拉斯算子的第一特征值,u(t)是相应于λ的特征函数,i.e.,-Δu=λu.那么沿着非规范化的Hk流,我们有相似地,沿着规范化的Hk流,有了这个定理之后我们可以很容易的得到以下推论
推论B：设Mn(t)为非规范化Hk流的解,今λ=λ(t)为Mn(t)上拉普拉斯算子的第一特征值,如果初始的二维曲面是全脐的,那么λ=λ(t)沿着非规范化的Hk流是非减的。
最近,双曲几何流受到了广泛的关注。在第三章的内容中,我们将要考虑由..教授和孔德兴教授引进的双曲Yambe流。从偏微分方程的角度来看,这是一个高度非线性双曲方程。我们在这一章中构造了这个方程的三类精确解。我们相信这些精确解对于研究这个方程的适定性以及其它一些基本的性质会有很大帮助。
第一类解是具有初始度量为Einstein的解。第二类解是具有轴对称的解。最后,作为这种流的特殊解,我们定义了稳定双曲Yamabe孤子,而且我们得到了这种孤子解所满足的方程。
以上是关于几何流部分的内容。下面两章是关于拓扑中一些问题的研究。
在第四章中,我们讨论了R3中完备定向极小曲面端的问题,我们给出了这个端的个数的一个上界,而且得到Hoffman和Meeks猜想在一定的特殊条件下是成立的。
Hoffman和Meeks猜想：令S是R3中完备定向极小嵌入曲面,满足∫s│K│<∞,那么r≤g+2,其中r为曲面S的端得个数,g为S紧化后的亏格。
定理C：令M是R3中完备定向极小曲面,满足∫s│K│<∞,而且M不是平面,那么它的端得个数r满足其中g是M紧化后的亏格,K为M的高斯曲率。
定理中的λ为下面的R3中紧域序列D上的特征值问题
紧致超曲面的刚性定理一直是一个很重要的课题,其中最引人注目的一个研究成果就是单位球面中紧致超曲面的数量曲率和平均曲率成比例时的刚性定理。
在第五章,我们主要讨论实欧式空间中以及Lorentz空间形式中类空紧致超曲面的刚性定理。具体来说,
定理E：令M是具有非负截曲率浸入到空间形式Nn+1(c)(c≥0)中的n维紧致超曲面。如果数量曲率r和平均曲率H满足r=f(H),这里f满足(n-1)(f')2-4nHf'+4nf-4nc≥0,那么M或者是全脐的,或者M=Sn-k×Sk。
同样地,我们可以把这个结果推广到Lorentz空间形式中的超曲面。
定理F：令M是具有非负截曲率的Lorentz空间形式R1n+1(c)(c>0)中的类空紧致超曲面,如果M的规范化的数量曲率r与平均曲率满足r=f(H),这里函数声f满足那么我们得到M是全脐曲面。
This paper mainly include two parts. The second and the third chapters belong to geometric flow part, and the last two chapters are topology part.
Many of the most exciting recent developments in geometric analysis have arisen from the study of geometric flow equation. Among the most prominent examples, one is the proof of the Riemannian Penrose inequality given by Huisken and Ilmanen by using the inverse mean curvature flow, and the other is the proof of Poincare conjecture given by Huaidong Cao and Xiping Zhu by using Ricci flow.
In the second chapter, we mainly study the first eigenvalue of Laplace op-erator along Hk-flow. Firstly, we derive the evolution equation for the first eigenvalue along Hk-flow.
Let F0:M→Rn+1 be a smooth immersion of an n-dimensional convex hypersurface in Euclidean space, where n≥2 and H(Fo(Mn))> 0. The evolution of M0=F0(M) by powers of mean curvature flow is the one-parameter family of smooth immersions F:M x [0,T)→Rn+1 satisfying where k>0, H is the mean curvature and v is the outer unit normal. We call the above system unnormalized Hk-flow. Note that for k=1, this flow is the mean curvature flow.
Theorem A:Assume Mn(t) is the solution of the Hk-flow. Letλ(t) be the first eigenvalue of Laplacian on Mn(t), and u(t) be the eigenfunction correspond-ing toλ,i.e.,-Δu=λu. If the metric evolves by the normalized Hk-flow, then we have Similarly, along the unnormalized Hk—low,
Therefore, we can get the following corollary
Corollary B:Letλ=λ(t) be the first eigenvalue of the Laplace operator on Mn(t) which evolves by the normalized Hk—flow. If the initial surface M is totally umbilical, then the eigenvalue is nondecreasing along the unnormalized Hk-flow.
Recently, hyperbolic geometric flow has received considerable attention.
In the third chapter, we consider the hyperbolic Yamabe flow introduced by Professor Kefeng Liu and Dexing Kong.
Let M be n-dimensional complete Riemannian manifold with Riemannian metric gij. Considering the following geometric flow
we derive some solutions for hyperbolic Yamabe flow. Firstly, the solutions of Einstein initial metric are given. Secondly, we investigate the solutions with axial symmetry. At last, as the special solution of the flow, the steady hyperbolic Yamabe soliton is defined and we get the equation satisfied by the soliton solution.
In the next two chapters, we will consider the topology part.
In the fourth chapter, we deal with the ends of the complete oriented minimal surface. We give an explicit upper bound for the number of the ends and derive Hoffman and Meeks'conjecture is true in special cases.
Hoffman and Meeks'conjecture:For complete minimal embedded surface S, assume S has finite total curvature, then g≥r-2, that is to say r≤g+2. In this theorem, r is the number of the ends of surface S and g is the genus of compactification of S.
Theorem C:Let M be a complete oriented minimal embedded surfaces in R3, which satisfies∫S|K|<∞, and it is not a plane, then the number of the ends r satisfies 2≤r≤2K-λ/4K+λ(g-1),g is the 9enus of compactification of M.
For A in the above theorem is the following eigenvalue form. where D is the compact domain sequence and K represents the Gauss curvature on M.
The topic on rigidity theorems of compact hypersurfaces has been studied extensively. One remarkable result of them is about compact hypersurfaces in a unit sphere with scalar curvature proportional to mean curvature.
In the last chapter, we have studied the rigidity theorems of compact hyper-surfaces in real space forms and compact spacelike hypersurfaces in Lorentzian space forms.
Theorem E:Let M be an n-dimensional compact hypersurface with non-negative sectional curvature in space forms Nn+1(c)(c≥0). If the normalized scalar curvature r and the mean curvature H satisfies r=f(H), where the function f satisfies (n-1)(f')2-4nHf'+4nf-4nc≥0, then M is either totally umbilical, or c>0, M=Sn-k×Sk.
Moreover, we consider the compact spacelike hypersurface in Lorentzian space forms. Similarly, we get the following rigidity theorems.
Theorem F:Let M be an n-dimensional compact spacelike hypersur-face with nonnegative sectional curvature in (n+1)-dimensional Lorentzian space forms R1n+1(c)(c> 0). Suppose the normalized scalar curvature r and mean curvature H satisfies r=f(H), where the function f satisfies then M is totally umbilical.
近期,几何分析中最重要的发展来自于对几何流方程的研究。其中,最重要的成果是：Huisken与Ilmanen用逆平均曲率流解决了黎曼Penrose不等式和曹怀东与朱熹平用Ricci流工具证明了庞卡莱猜想。
在第二章,我们主要研究了Hk流下拉普拉斯算子的第一特征值,我们首先得到了这个流下拉普拉斯算子第一特征值的发展方程。
我们考虑Rn+1中的紧致无边的,凸的n维超曲面M。令F0:M→Rn+1是到欧式空间的光滑浸入,这里n≥2且H(F0(Mn))>0.M0=F0(M)在Hk流下的演化是一个光滑浸入的单参数族F：M×[0,T]→Rn+1,它满足这里k>0,H是平均曲率且v是外法线方向。我们称上述系统为非规范化的Hk流。注意到k=1,这种流就是非规范后的平均曲率流。
定理A：设Mn(t)为非规范化Hk流的解,令λ=λ(t)为Mn(t)上拉普拉斯算子的第一特征值,u(t)是相应于λ的特征函数,i.e.,-Δu=λu.那么沿着非规范化的Hk流,我们有相似地,沿着规范化的Hk流,有了这个定理之后我们可以很容易的得到以下推论
推论B：设Mn(t)为非规范化Hk流的解,今λ=λ(t)为Mn(t)上拉普拉斯算子的第一特征值,如果初始的二维曲面是全脐的,那么λ=λ(t)沿着非规范化的Hk流是非减的。
最近,双曲几何流受到了广泛的关注。在第三章的内容中,我们将要考虑由..教授和孔德兴教授引进的双曲Yambe流。从偏微分方程的角度来看,这是一个高度非线性双曲方程。我们在这一章中构造了这个方程的三类精确解。我们相信这些精确解对于研究这个方程的适定性以及其它一些基本的性质会有很大帮助。
第一类解是具有初始度量为Einstein的解。第二类解是具有轴对称的解。最后,作为这种流的特殊解,我们定义了稳定双曲Yamabe孤子,而且我们得到了这种孤子解所满足的方程。
以上是关于几何流部分的内容。下面两章是关于拓扑中一些问题的研究。
在第四章中,我们讨论了R3中完备定向极小曲面端的问题,我们给出了这个端的个数的一个上界,而且得到Hoffman和Meeks猜想在一定的特殊条件下是成立的。
Hoffman和Meeks猜想：令S是R3中完备定向极小嵌入曲面,满足∫s│K│<∞,那么r≤g+2,其中r为曲面S的端得个数,g为S紧化后的亏格。
定理C：令M是R3中完备定向极小曲面,满足∫s│K│<∞,而且M不是平面,那么它的端得个数r满足其中g是M紧化后的亏格,K为M的高斯曲率。
定理中的λ为下面的R3中紧域序列D上的特征值问题
紧致超曲面的刚性定理一直是一个很重要的课题,其中最引人注目的一个研究成果就是单位球面中紧致超曲面的数量曲率和平均曲率成比例时的刚性定理。
在第五章,我们主要讨论实欧式空间中以及Lorentz空间形式中类空紧致超曲面的刚性定理。具体来说,
定理E：令M是具有非负截曲率浸入到空间形式Nn+1(c)(c≥0)中的n维紧致超曲面。如果数量曲率r和平均曲率H满足r=f(H),这里f满足(n-1)(f')2-4nHf'+4nf-4nc≥0,那么M或者是全脐的,或者M=Sn-k×Sk。
同样地,我们可以把这个结果推广到Lorentz空间形式中的超曲面。
定理F：令M是具有非负截曲率的Lorentz空间形式R1n+1(c)(c>0)中的类空紧致超曲面,如果M的规范化的数量曲率r与平均曲率满足r=f(H),这里函数声f满足那么我们得到M是全脐曲面。
This paper mainly include two parts. The second and the third chapters belong to geometric flow part, and the last two chapters are topology part.
Many of the most exciting recent developments in geometric analysis have arisen from the study of geometric flow equation. Among the most prominent examples, one is the proof of the Riemannian Penrose inequality given by Huisken and Ilmanen by using the inverse mean curvature flow, and the other is the proof of Poincare conjecture given by Huaidong Cao and Xiping Zhu by using Ricci flow.
In the second chapter, we mainly study the first eigenvalue of Laplace op-erator along Hk-flow. Firstly, we derive the evolution equation for the first eigenvalue along Hk-flow.
Let F0:M→Rn+1 be a smooth immersion of an n-dimensional convex hypersurface in Euclidean space, where n≥2 and H(Fo(Mn))> 0. The evolution of M0=F0(M) by powers of mean curvature flow is the one-parameter family of smooth immersions F:M x [0,T)→Rn+1 satisfying where k>0, H is the mean curvature and v is the outer unit normal. We call the above system unnormalized Hk-flow. Note that for k=1, this flow is the mean curvature flow.
Theorem A:Assume Mn(t) is the solution of the Hk-flow. Letλ(t) be the first eigenvalue of Laplacian on Mn(t), and u(t) be the eigenfunction correspond-ing toλ,i.e.,-Δu=λu. If the metric evolves by the normalized Hk-flow, then we have Similarly, along the unnormalized Hk—low,
Therefore, we can get the following corollary
Corollary B:Letλ=λ(t) be the first eigenvalue of the Laplace operator on Mn(t) which evolves by the normalized Hk—flow. If the initial surface M is totally umbilical, then the eigenvalue is nondecreasing along the unnormalized Hk-flow.
Recently, hyperbolic geometric flow has received considerable attention.
In the third chapter, we consider the hyperbolic Yamabe flow introduced by Professor Kefeng Liu and Dexing Kong.
Let M be n-dimensional complete Riemannian manifold with Riemannian metric gij. Considering the following geometric flow
we derive some solutions for hyperbolic Yamabe flow. Firstly, the solutions of Einstein initial metric are given. Secondly, we investigate the solutions with axial symmetry. At last, as the special solution of the flow, the steady hyperbolic Yamabe soliton is defined and we get the equation satisfied by the soliton solution.
In the next two chapters, we will consider the topology part.
In the fourth chapter, we deal with the ends of the complete oriented minimal surface. We give an explicit upper bound for the number of the ends and derive Hoffman and Meeks'conjecture is true in special cases.
Hoffman and Meeks'conjecture:For complete minimal embedded surface S, assume S has finite total curvature, then g≥r-2, that is to say r≤g+2. In this theorem, r is the number of the ends of surface S and g is the genus of compactification of S.
Theorem C:Let M be a complete oriented minimal embedded surfaces in R3, which satisfies∫S|K|<∞, and it is not a plane, then the number of the ends r satisfies 2≤r≤2K-λ/4K+λ(g-1),g is the 9enus of compactification of M.
For A in the above theorem is the following eigenvalue form. where D is the compact domain sequence and K represents the Gauss curvature on M.
The topic on rigidity theorems of compact hypersurfaces has been studied extensively. One remarkable result of them is about compact hypersurfaces in a unit sphere with scalar curvature proportional to mean curvature.
In the last chapter, we have studied the rigidity theorems of compact hyper-surfaces in real space forms and compact spacelike hypersurfaces in Lorentzian space forms.
Theorem E:Let M be an n-dimensional compact hypersurface with non-negative sectional curvature in space forms Nn+1(c)(c≥0). If the normalized scalar curvature r and the mean curvature H satisfies r=f(H), where the function f satisfies (n-1)(f')2-4nHf'+4nf-4nc≥0, then M is either totally umbilical, or c>0, M=Sn-k×Sk.
Moreover, we consider the compact spacelike hypersurface in Lorentzian space forms. Similarly, we get the following rigidity theorems.
Theorem F:Let M be an n-dimensional compact spacelike hypersur-face with nonnegative sectional curvature in (n+1)-dimensional Lorentzian space forms R1n+1(c)(c> 0). Suppose the normalized scalar curvature r and mean curvature H satisfies r=f(H), where the function f satisfies then M is totally umbilical.
引文
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[9]W. R. Dai, D. X. Kong, and K. F. Liu. Hyperbolic geometry flow (Ⅰ):short-time existence and nonlinear stability. Pure and Applied Mathematics Quar-terly. Special Issue Dedicated to M. Atiyah's 80th Birthirday. To appear.
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[2]X.D.Cao, Eigenvalues of (-Δ+R/2) on manifolds with nonnegative curvature operator. Math. Ann, Vol 337 (2),(2007),435-441.
[3]S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann.255(1997),195-204.
[4]S. Y. Cheng and S. T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Math. Ann.104(1976),407-419.
[5]Q. M. Cheng, Hypersurfaces in a unit sphere Sn+1 with constant scalar curvature, J. London Math. Soc64 (2)(2001),755-768.
[6]D.Fischer-Colbrie and R. Shoen, The structure of complete stable minimal surfaces in three-manifolds of non-negative scalar curvature, Comm. Pure. Appl. Math,33, (1980),199-211.
[7]T. H. Colding and W. P. Minicozzi Ⅱ, Embedded minimal disks:proper versus nonproper-global versus local, Transactions of A.M.S.,356(1)(2003), 283-289.
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