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。     

常曲率黎曼流形中的一类超曲面

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)。     

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.     

S~4中具有调和共形高斯映照的超曲面

设x:M~n→S~(n+1)是球面S~(n+1)中的一个定向超曲面,其共形高斯映照G=(H,Hx+en+.1):M~n→R_1S~(n+3)是M(o|¨)bius变换群下的一个不变量,其中H,e(n+1)+1分别是超曲面x的平均曲率和单位法向量场.本文研究了S~4中具有调和共形高斯映照的超曲面,分类了具有调和共形高斯映照和常M(o|¨)bius数量曲率的超曲面,给出了具有调和共形高斯映照但不是Willmore超曲面的例子.     

单位球面中具平行单位平均曲率向量的子流形

设M是n-维闭黎曼流形,等距浸入(n+p)-维单位球空间S^n+p,具有平行的单位平均曲率向量。若S≤min{2n/3,2(n-1)^1/2},其中S是M的第二基本形式长度的平方,则M是S^n+p的一个(n+1)-维全测地子流形Sⁿ⁺¹中的超曲面。     

拟常曲率空间中具有常平均曲率的完备超曲面

主要研究了拟常曲率空间中具有常平均曲率的完备超曲面,得到了这类超曲面全脐的一个结果.即若Nn+1的生成元η∈TM,且a-2|b|=c(常数)>0,则当S<2 n-1~(1/2)(a-2|b|)时,M为全脐超曲面.     

有界F-支撑函数和类空Wulff形

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.     

Lorentz流形中的类空超曲面

1.引言在[1]中,Calabi证明了n+1(n≤4)维Minkowski空间中的完备极大类空超曲面是全测地的。在[2]中 , Cheng-Yau对所有的n证明了这一结论。在[3]中,对于某一类Lorentz流形,Nishikawa证明了类似的结果。并且在[2]中,Cheng-Yau还证明了当具有常数平均曲率的类空超曲面M是Minkowski空间的闭子集时,有     

关于拟常曲率流形的子流形的Simons型公式

对于浸入在常曲率流形中的超曲面,K.Nomizu and B.Smyth在[1]中计算其第二基本张量的长度平方的拉氏算子得到一个Simons型公式,运用这个公式,他们研究了在一些附加条件下R~(n 1)或S~(n 1)中超曲面的测定。J.Erbacher[2]和K.Yano and S.Ishihara[3]把[1]的结果推广到浸入在常曲率流形中余维为p(≥1)的子流形上去。本文把[2,3]的Simons型公式推广到拟常曲率流形的情形,用此公式我们求得拟常曲率流形的极小子流形为全测地子流形的一个充分条件,还给出这个公式的其他一些应用。     

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;     

Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds

Our purpose in this paper is to study the rigidity of complete linear Weingarten hypersurfaces immersed in a locally symmetric manifold obeying some standard curvature conditions (in particular, in a Riemannian space with constant sectional curvature). Under appropriated constrains on the scalar curvature function, we prove that such a hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. Furthermore, we also deal with the parabolicity of these hypersurfaces with respect to a suitable Cheng–Yau modified operator.     

Some Theorems for Hypersurface of Randers Spaces

In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. (1) Let (Mⁿ⁺¹, F) be a Randers-Minkowski space. If (Mⁿ, F) is a hypersurface of (Mⁿ⁺¹, F) with constant flag curvature K=1, then we can prove that M is Riemannian. (2) Let (Mⁿ⁺¹, F) be a Randers space with constant flag curvature. Assume (M, F) is a compact hypersurface of (Mⁿ⁺¹, F) with constant mean curvature|H|. Then a pinching theorem is established, which generalizes the result of[Proc. Amer. Math. Soc., 120, 1223-1229 (1994)] from the Riemannian case to the Randers space.     

A maximum principle for hypersurfaces with constant scalar curvature and applications

In this article, we establish a weak maximum principle for complete hypersurfaces with constant scalar curvature into Riemannian space forms, and give some applications to estimate the norm of the traceless part of its second fundamental form.     

On the Gauss mapping of hypersurfaces with constant scalar curvature in ℍ n+1

In this paper, we deal with complete hypersurfaces immersed in the hyperbolic space with constant scalar curvature. By supposing suitable restrictions on the Gauss mapping of such hypersurfaces we obtain some rigidity results. Our approach is based on the use of a generalized maximum principle, which can be seen as a sort of extension to complete (noncompact) Riemannian manifolds of the classical Hopf’s maximum principle.     

Hypersurfaces with constant anisotropic mean curvature in Riemannian manifolds

We formulate a variational notion of anisotropic mean curvature for immersed hypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropic mean curvature are characterized as critical points of an elliptic parametric functional subject to a volume constraint. We provide examples of such hypersurfaces in the case of rotationally invariant functionals defined in product spaces. These examples include rotationally invariant hypersurfaces and graphs.     

Local imbedding of Einstein spaces

Summary A special class of hypersurfaces of a Riemannian space is examined, this class being defined by the stipulation that the coefficients of the third fundamental form be expressible as linear combinations of the coefficients of the first and second fundamental forms. It is jound that these so-called C-hypersurfaces are umbilical provided that certain conditions (which may depend on dimension) are satisfied. An (n-1)-dimensional Einstein space imbedded in an n-dimensional space of constant curvature is such a C-hypersurface; accordingly the theory may be applied to the problem of the local imbedding of such spaces. Entrata in Redazione il 23 giugno 1971.     

Reduction of the n-dimensional static vacuum Einstein equation and generalized Schwarzschild solutions

We consider the static vacuum Einstein spacetime when the spatial factor is conformal to a n-dimensional pseudo-Euclidean space. The most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described. We obtain the entire set of solutions of the reduced system, where the classical Schwarzschild solution arises as a particular solution. In addition, we show that the Riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature.     

Stable minimal hypersurfaces in a Riemannian manifold with pinched negative sectional curvature

We give an estimate of the smallest spectral value of the Laplace operator on a complete noncompact stable minimal hypersurface M in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface M has sufficiently small total scalar curvature then M has only one end. We also obtain a vanishing theorem for L ² harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover, we provide sufficient conditions for a minimal hypersurface in a Riemannian manifold with nonpositive sectional curvature to be stable.     

Eigenvalue estimate on complete noncompact Riemannian manifolds and applications

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

     

On hypersurfaces with two distinct principal curvatures in space forms

We investigate the immersed hypersurfaces in space forms ℕ ^n + 1(c), n ≥ 4 with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant. We prove that any immersed hypersurface in space forms with two distinct non-simple principal curvatures is locally conformal to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for the Clifford hypersurfaces in terms of the trace free part of the second fundamental form.