空间形式中紧致超曲面的刚性

设(M~n,g)是一个黎曼流形,f:M~n→Q~(n+1)(c)是一个等距浸入,其中Q~(n+1)(c)是n+1维的空间形式.如果对于任一个等距浸入f:M~n→Q~(n+1)(c),都存在等距变换φ:Q~(n+1)(c)→Q~(n+1)(c),使得φ·f=f,则称f(M~n)具有刚性.本文证明:如果超曲面是紧致的,(1)当c≤0时,如果紧致超曲面的维数大于或等于3,则紧致超曲面具有刚性;(2)当c0时,如果紧致超曲面的维数大于或等于5,则空间形式中紧致超曲面具有刚性;这推广了经典的Cohn-Vossen定理.     

球面S~(n+1)(1)中的紧致2-调和超曲面

本文得到了S~(n+1)(1)中2-调和超曲面的一些结果.首先,我们将J.Simons的Pinching定理推广到2-调和超曲面上.当n=2,3时,我们还给出了它们的分类;其次,我们证明了S~3(1)中常平均曲率曲面的Pinching定理并得到了它们的分类;最后,我们给出了S~(n+1)(1)(n≤10)中具有非负截曲率的2-调和超曲面的分类;     

圆球面为边的常F-平均曲率类空超曲面

设F(t)是上伪球面H_+~n上时轴旋转不变且满足凸条件的光滑函数.本文证得:Lorentz空间L~(n+1)中,以圆球面S~(n-1)(R)为边界且有非零常F-平均曲率的紧致类空超曲面必为Wulff帽.     

正曲率空间形式中超曲面的全脐性与高阶平均曲率

本文研究正曲率空间形式S~(n+1)(c)(c0)中紧致的闭的等距浸入超曲面M~n的全脐性质和高阶平均曲率,所得结果改进和推广了这方面最近有关定理.     

单位球面中紧致超曲面的曲率结构与拓扑性质

设M是单位球面S~(n 1)(1)中的n维(n■3)紧致连通定向超曲面,本文研究这种超曲面的曲率结构与拓扑性质,利用Lawson和Simons关于稳定k维流的不存在性与同调群消失定理,得到了曲率与拓扑的一个关系定理,从而对Cheng Q.M.所提出的一个分类问题从拓扑角度给出了一个肯定回答,并且部分肯定回答了Cheng的另一个问题.     

S~(n+1)(1)内凸超曲面为边界的区域的第一特征值估计

Ｍ是Ｓｎ＋１(１)内紧致嵌入凸超曲面．Ｍ分Ｓｎ＋１(１)为两个连通区域Ω１和Ω２．Ω１＝Ｍ和Ω１是凸的．本文估计了Ω１的Ｌａｐｌａｃｅ算子的第一特征值的下界．     

常曲率黎曼流形的共形平坦的极小超曲面(英文)

本文研究常曲率黎曼流形 S~(n+1)(c)中的共形平坦的极小超曲面 M~h,证明了下面结果.定理 设 M~h 是 n+1维常曲率黎曼流形 S~(n+1)(c)的共形平坦超曲面(n≥4),则 M~n是常数量曲率的极小超曲面的充要条件是:(1)M~n 的数量曲率 R=(n-1)c 时,M~n 是全测地超曲面,从而也有常曲率 c;(2)M~n 的数量曲率 R≠n(n-1)c 时,c>0和 M~n 局部可约为常曲率黎曼流形S~(n-1)(n/(n-1) c)与直线 R′的乘积.系,设 M~n 是具有非正常曲率 c 的黎曼流形 S~(n+1)(c)的共形平坦超曲面(n≥4),如果M~n 是常数量曲率的极小超曲面,则 M~n 是全测地超曲面。     

anti-de Sitter 空间中紧致类空超曲面的积分公式及其在常高阶平均曲率下的应用

该文对 anti-de Sitter 空间H₁ⁿ⁺¹中的紧致类空超曲面建立了积分公式,并应用它们在常高阶平均曲率的条件下讨论了H₁ⁿ⁺¹中紧致类空超曲面的全脐问题.     

球空间S^n＋p（C）中的紧致极小子流形

设 M~n 是常曲率空间 S~(n+p)(c)的紧致极小子流形,设 K 和 Q 分别是 M~n 上每点各方向截面曲率和 Ricci 曲率的下确界,R 是 M~n 的数量曲率,本文利用 M~n 的内在量 KQ和 R,给出球空间中紧致极小子流形是全测地子流形的六个充分条件。     

E_(s)^(5)中的双调和超曲面北大核心CSCD

本文研究五维伪欧氏空间E_(s)^(5)中的双调和超曲面的几何和分类问题,证明了:如果M^(4)_(r)是E_(s)^(5)(s=1,2,3,4)中具有对角化形状算子的双调和超曲面,那么M^(4)_(r)一定是极小的.结合Turgay等人结果,本文进一步表明了五维Minkowski空间E_(1)^(5)中Lorentz双调和超曲面一定是极小的.该结果证明了五维伪欧氏空间中超曲面情形下的双调和猜想.     

Sn+1(1)中的极小超曲面的等谱问题

设M为Sn 1(1)中紧致极小超面Mn1,n2= Sn1nn1×Sn2nn2 Sn 1(1)为Sn 1(1)中的Clifford极小超曲面如果Specp( M) =specp( Mn1,n2) ,Specq( M) =specq( Mn1,n2) ,其中0≤p 相似文献   

First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere

S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ ₁ of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S ⁿ⁺¹. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S ⁿ⁺¹ with λ ₁ = n is either isometric to the sphere S ⁿ or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate.     

A note on hypersurfaces in a sphere

We study the influence of a unit Killing vector field on the geometry of a hypersurface in the unit sphere. The combination of the Killing vector field on the hypersurface and the conformal vector field on the ambient sphere triggers the presence of four specific smooth functions on the hypersurface, we use these four functions to derive different sufficient conditions for a hypersurface to be the totally geodesic sphere and for a minimal hypersurface to be the totally geodesic sphere, Clifford minimal hypersurface respectively. In particular we classify compact minimal hypersurfaces with a unit Killing vector field in the unit sphere.     

On space-like hypersurfaces in the de Sitter space

This paper gives the intrinsic conditions for a compact space-like hypersurface in a de Sitter space to be isometric to a sphere.     

Minimal Surfaces in a Sphere and the Ricci Condition

In this paper we deal with minimal surfaces in a sphere which are locally isometric to a minimal surface in S3. We prove that a minimal surface in a sphere is locally isometric to a minimal surface in S3 if the curvature ellipse has constant and positive eccentricity. Moreover, we prove the following rigidity result: a compact minimal surface M in Sm, m 6, cannot be locally isometric to a minimal surface in S3 unless M already lies in S3 or M is flat and lies in S5.     

de Sitter 空间中的常平均曲率完备类空超曲面

设Ｍ是常曲率ｃ的ｄｅ Ｓｉｔｔｅｒ空间Ｓ１＾ｎ＋１（ｃ）的常平均曲率的完备类空超曲面,Ｓ表示第二形式的范数平方。本文证明：差Ｓ〈２√ｎ－１ｃ,则Ｍ是全脐的和等距于一球面。     

First nonzero eigenvalue of a minimal hypersurface in the unit sphere

In this paper, it is shown that the first nonzero eigenvalue λ₁ of the Laplacian operator on a compact immersed minimal hypersurface M in the unit sphere S ⁿ⁺¹ satisfies one of the following $$ (i)\lambda _{1}=n, \quad (ii)\lambda _{1} \leq (1+k_{0})n, \quad (iii)\lambda _{1}\geq n+\frac{n}{2}(nk_{0}-(n-1))$$ where k ₀ is the infimum of the sectional curvatures of M. It is also shown that a compact immersed minimal hypersurface of the unit sphere S ⁿ⁺¹ with λ₁?=?n is either isometric to the unit sphere S ⁿ or else k ₀?<?n ^?1(n?1).     

Hypersurfaces of the Euclidean sphere with nonnegative Ricci curvature

In this paper we prove that a compact oriented hypersurface of a Euclidean sphere with nonnegative Ricci curvature and infinite fundamental group is isometric to an H(r)-torus with constant mean curvature. Furthermore, we generalize, whithout any hypothesis about the mean curvature, a characterization of Clifford torus due to Hasanis and Vlachos. Received: 19 March 2002     

De Sitter空间中具有非负截曲率的常平均曲率完备类空超曲面

设M为de Sitter空间S1^n 1（1）中的完备（非紧）类空超曲面，具有常平均曲率和非负截曲率，在适当条件下，我们证明了它与欧式空间或者双曲柱面等距。     

Clifford 环面 Sm(\sqrt{\frac{m}{n}})×Sn-m(\sqrt{\frac{n-m}{n}})的刚性定理

\small\zihao{-5}\begin{quote}{\heiti 摘要:} 设$M$为$n+1$维单位球面$S^{n+1}(1)$中的一个极小闭超曲面，如果 $ n \le S \le n+\frac{2}{3}$, 则有 $S=n$ 且 $M$ 与某一Clifford 环面 $S^m(\sqrt{m/n}) \times S^{n-m}(\sqrt{(n-m)/n})$等距.