The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ ...The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ λj＋ nH）,S = ∑i（ λi＋ nH）~2, where h（ij）= λiδ（ij）. Utilizing Lagrange＇s method, a sharper pointwise estimation of 3（A- 2B） in terms of S and ｜▽h｜~2 is obtained, here ｜▽h｜~2= ∑i,j,k h（ijk）~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.展开更多
Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n）...Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n） and Ф o x（p） = x（q）, then the hypersurface is called a Mobius homogeneous hypersurface. In this paper, the Mobius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mobius transformation.展开更多
An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. In this paper, we give a complete clas...An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. In this paper, we give a complete classification for all Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues.展开更多
In this paper,the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied.It is proved that the flow converges to a unique geodesi...In this paper,the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied.It is proved that the flow converges to a unique geodesic sphere,i.e.,every principle curvature of the hypersurfaces converges to a same constant under the flow.展开更多
文摘The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ λj＋ nH）,S = ∑i（ λi＋ nH）~2, where h（ij）= λiδ（ij）. Utilizing Lagrange＇s method, a sharper pointwise estimation of 3（A- 2B） in terms of S and ｜▽h｜~2 is obtained, here ｜▽h｜~2= ∑i,j,k h（ijk）~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.
基金supported by the National Natural Science Foundation of China(Nos.11571037,11471021)
文摘Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n） and Ф o x（p） = x（q）, then the hypersurface is called a Mobius homogeneous hypersurface. In this paper, the Mobius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mobius transformation.
基金supported by National Natural Science Foundation of China (Grant Nos. 10671181, 11071225)
文摘An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. In this paper, we give a complete classification for all Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues.
文摘In this paper,the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied.It is proved that the flow converges to a unique geodesic sphere,i.e.,every principle curvature of the hypersurfaces converges to a same constant under the flow.
