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1.
刘艳楠  简怀玉 《中国科学A辑》2006,36(12):1404-1412
考虑由平均曲率和外力场之差支配的超曲面的发展运动. 证明了如果初始曲面的平均曲率大于某一个仅依赖于外力场导数的常数, 则这样的流将在有限时间爆破. 对于线性外力, 可以证明在运动过程中超曲面的凸性是保持的, 并且如果初始曲率小于某一与外力场有关的常数, 则流将光滑的存在于任意有限时间, 且曲面将扩张到无穷.  相似文献   

2.
考虑欧氏空间R~(n+p)中n(≥2)维闭子流形沿平均曲率向量场加上一个位置向量方向外力场的流的发展.设子流形任意一点处平均曲率向量非零和第二基本形式的模长以平均曲率向量长度的常数倍(仅与n有关)为界,我们证明了若外力场很小时,拼挤子流形要么在有限时间内收缩为一点,要么子流形在任意时刻都存在;若外力场足够大时,子流形发散到无穷大;同时,当流发展到极限位置时,规范化子流形都光滑收敛到R~(n+p)中的一个n+1维子空间中的标准球面.  相似文献   

3.
若超曲面的Laguerre形式为零且Laguerre第二基本形式的特征值(称为Laguerre主曲率)为常数,则称超曲面为Laguerre等参超曲面.对R~6中的Laguerre等参超曲面进行了研究,得到了分类定理.  相似文献   

4.
设M~n是n+1维常由率黎曼流形S~(n+1)中的超曲面,其二个主曲率的重数L_1,L_2(L_1+L_2=n)保持为常数。本文证得:1.若L_1,L_2≥2则局部地至少有一个主曲率为常数。2.若L_1,L_2≥2,且M~n是常平均由率的单连通完备超曲面,则M~n=S~(L_1)×S~(L_2)。3.若L_1=1,L_2=n-1且M~n为常数量曲率和常平均曲率的单连通完备超曲面,则M~n=S~1×S~(n-1)。4.若M~n为单连通完备的S-流形,则 M~n=S~(L_1)×S~(L_2)。  相似文献   

5.
蔡开仁 《数学杂志》1998,18(2):139-149
本文证明了一个拼嵌的爱因斯坦流形中的任何超曲面在沿其平均曲率向量演化时,如果初发始曲面满足保持其截曲率为正的某些条件,则在有限时间内超曲而将收缩成一点。  相似文献   

6.
本文证明了下述的主要结果:设 M 为 R~4中的完备常中曲率超曲面,且数量曲率为常数,如果 M 不是常主曲率超曲面,则一定有R<0.  相似文献   

7.
张远征 《数学学报》2016,59(1):37-46
给定H_+~n上适合凸条件的正函数F,对L~(n+1)中具有非退化Gauss映射的类空超曲面引入了Θ_F曲率.对适当的F,本文证得:具有常Θ_F曲率,且F-支撑函数介于两个负常数之间的类空超曲面必是类空Wulff形.在F=1的情况下,对H_i/H_n为常数的类空超曲面也建立了类似的唯一性结果.  相似文献   

8.
本文是关于Riemann流形中超曲面逆曲率流的综述文章.首先介绍Euclid空间超曲面的逆曲率流的收敛性,以及其在证明Alexandrov-Fenchel不等式中的应用.其次,介绍在双曲空间以及球面中类似的结论.接着讨论Kottler空间的逆平均曲率流.Kottler空间是一类扭曲乘积空间,它满足物理中的稳态方程且在无穷远处渐近于局部双曲空间.本文将介绍此类空间中的逆平均曲率流的收敛性并用来对星形平均凸超曲面证明Minkowski型不等式.逆曲率流是近几年比较热门的一个研究领域,然而,由于篇幅有限,本文不能一一全部介绍.因此,本文最后列举一些相关的文献供感兴趣的读者参考.  相似文献   

9.
本文证明了下述的主要结果: 设M为R~4中的完备常中曲率超曲面,且数量曲率为常数,如果M不是常主曲率超曲面,则一定有R<0。  相似文献   

10.
许志才 《数学杂志》1998,18(4):466-468
设M^n是De Sitter空间S1^n+1中具有常数平均曲率且第二基本形式长度的平方为常数的完备类空超曲面,若S≤2(n-1)^1/2,则M^n是等参超曲面。  相似文献   

11.
In this paper, we study the evolution of hypersurface moving by the mean curvature minus an external force field. It is shown that the flow will blow up in a finite time if the mean curvature of the initial surface is larger than some constant depending on the boundness of derivatives of the external force field. For a linear force, we prove that the convexity of the hypersurface is preserved during the evolution and the flow has a unique smooth solution in any finite time and expands to infinity as the time tends to infinity if the initial curvature is smaller than the slope of the force.  相似文献   

12.
A smooth, compact and strictly convex hypersurface evolving in ℝ n+1 along its mean curvature vector plus a forcing term in the direction of its position vector is studied in this paper. We show that the convexity is preserving as the case of mean curvature flow, and the evolving convex hypersurfaces may shrink to a point in finite time if the forcing term is small, or exist for all time and expand to infinity if it is large enough. The flow can converge to a round sphere if the forcing term satisfies suitable conditions which will be given in the paper. Long-time existence and convergence of normalization of the flow are also investigated.  相似文献   

13.
In this paper, we study the evolution of a noncompact hypersurface moving by mean curvature minus an external force field. We prove that the flow has a long-time smooth solution for a kind of special external force fields if the initial hypersurface is a Lipschitz entire graph with linear growth.  相似文献   

14.
In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.   相似文献   

15.
This paper is concerned with the following three types of geometric evolution equations: the volume preserving mean curvature flow, the intermediate surface diffusion flow, and the surface diffusion flow. Important common properties of these flows are the preservation of volume and the decrease of perimeter. It is shown in this paper that the intermediate surface diffusion flow can lose convexity. Hence the volume preserving mean curvature flow is the only flow among the evolution equations under consideration which preserves convexity, cf. [11, 16, 14, 17]. Moreover, several sufficient conditions are presented, which illustrate that each of the above mentioned flows can move smooth initial configurations into singularities in finite time.  相似文献   

16.
研究由仿射平均曲率支配的严格凸超曲面的发展运动.在假定仿射平均曲率流存在并且曲面保持严格凸的条件下,通过对曲面支撑函数的计算,给出了高斯曲率的发展方程.  相似文献   

17.
We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.  相似文献   

18.
We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main driving factor, we refer to this model as the hyperbolic mean curvature flow (HMCF). The case that the initial velocity field is normal to the hypersurface is of particular interest: this property is preserved during the evolution and gives rise to a comparatively simpler evolution equation. We also consider the case where the manifold can be viewed as a graph over a fixed manifold. Our main results are as follows. First, we derive several balance laws satisfied by the hypersurface during the evolution. Second, we establish that the initial-value problem is locally well-posed in Sobolev spaces; this is achieved by exhibiting a convexity property satisfied by the energy density which is naturally associated with the flow. Third, we provide some criteria ensuring that the flow will blow-up in finite time. Fourth, in the case of graphs, we introduce a concept of weak solutions suitably restricted by an entropy inequality, and we prove that a classical solution is unique in the larger class of entropy solutions. In the special case of one-dimensional graphs, a global-in-time existence result is established.  相似文献   

19.
Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special case of Finsler manifolds, we prove the short time existence and uniqueness for solutions of the mean curvature flow and prove that the flow preserves the convexity and mean convexity. We also derive some comparison principles for the mean curvature flow.  相似文献   

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