单位球面内常平均曲率超曲面的间隙性
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摘要
本文主要研究了单位球面内常平均曲率超曲面的第二基本形式模长平方的间隙性.
关于极小子流形的间隙问题已经有了很多好的结果,第一节引言中我们将介绍本文的研究背景及这些主要结果,第二节中我们来介绍具有常平均曲率情形的间隙问题的发展现状,在第四节中我们用类似的方法讨论平均曲率为常数(不一定是零)的紧致超曲面的第二间隙问题，得到了第二间隙δ=(?),此处当H=0时,我们有δ=2/3,文献[13]中曾给出了一个错误结论,本节中我们将摘录其证明过程并进行解析.第五节中我们还初步探索了第三间隙,得到了(?).但是考虑到这个第三间隙值的形式很复杂,我们在本节中将对其作进一步的改进,在补充条件的前提下得到了更简洁的间隙值(?).
为了证明定理,在第三节中我们将给出一些定义及重要的公式.
In this paper, we mainly study the gap of the second fundamental form of hypersurface with constant mean curvature in the unit sphere.
There have been many good results when M is minimal sub manifold. In the first section we will introduce the context and make a general description on the recent researches. In the second we will introduce the complexion of the development for this problem. In the fourth we'll discuss the second gap on special conditions using similar methods when the mean curvature is constant (maybe notzero), and find the second gap (?). Here when H=0,we haveδ=2/3. And a false claim was given in [13], we will cite his sketch of theproof pointing out his mistake. In the last section we'll also explore the third gap(?). Whereas theform of the third gap is particularly complex, we'll improve it in this section, andget the more sententious value (?) under the additional conditions.
We'll give some definitions and formulas in the third section to prove the theorems.
关于极小子流形的间隙问题已经有了很多好的结果,第一节引言中我们将介绍本文的研究背景及这些主要结果,第二节中我们来介绍具有常平均曲率情形的间隙问题的发展现状,在第四节中我们用类似的方法讨论平均曲率为常数(不一定是零)的紧致超曲面的第二间隙问题，得到了第二间隙δ=(?),此处当H=0时,我们有δ=2/3,文献[13]中曾给出了一个错误结论,本节中我们将摘录其证明过程并进行解析.第五节中我们还初步探索了第三间隙,得到了(?).但是考虑到这个第三间隙值的形式很复杂,我们在本节中将对其作进一步的改进,在补充条件的前提下得到了更简洁的间隙值(?).
为了证明定理,在第三节中我们将给出一些定义及重要的公式.
In this paper, we mainly study the gap of the second fundamental form of hypersurface with constant mean curvature in the unit sphere.
There have been many good results when M is minimal sub manifold. In the first section we will introduce the context and make a general description on the recent researches. In the second we will introduce the complexion of the development for this problem. In the fourth we'll discuss the second gap on special conditions using similar methods when the mean curvature is constant (maybe notzero), and find the second gap (?). Here when H=0,we haveδ=2/3. And a false claim was given in [13], we will cite his sketch of theproof pointing out his mistake. In the last section we'll also explore the third gap(?). Whereas theform of the third gap is particularly complex, we'll improve it in this section, andget the more sententious value (?) under the additional conditions.
We'll give some definitions and formulas in the third section to prove the theorems.
引文
[1] S. S. Chern, Docamo, S. Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length [J]. Func. Anal. and Rela. Fiel.Springer-Verlag, 1970: 59-75.
[2] C. K. Peng, C. L. Terng. Minimal hypersurface of spheres with constant scalar curvature [J]. Ann. Math., 1983, 226: 177-198.
[3] C. K. Peng, C. L. Terng. The scalar curvature of minimal hypersurfaces in spheres [J]. Ann. Math., 1983, 226: 105-113.
[4] H. C. Yang, Q. M. Cheng. A note on the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere [J]. Chinise Science Bull, 1991, 36: 1-6.
[5] H. C. Yang, Q. M. Cheng. An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit spheres [J]. Manuscripta. Math., 1994, 82: 89-100.
[6] H. C. Yang, Q. M. Cheng. Chern's conjecture on minimal hypersurfaces [J].Math. Z., 1998, 227: 377-390.
[7] Q. M. Cheng. The rigidity of Clifford torus (?)[J]. Comment Math Helvetici. 1996, 71: 60-69.
[8] 魏嗣明,许洪伟.数量曲率的第二间隙[J].浙江大学学报(理学版),2007,34(4):375-377.
[9] 孙自琪.S~4内的常数量曲率的紧致超曲面[J].数学年刊(A),1987,8(3):273-286.
[10] 许洪伟.高维常平均曲率超曲面的数量曲率的间隙[J].高校应用数学学报(A),1993,8(4):410-419.
[11] 纪永强.子流形几何[M].北京:科学出版社,2004,208-211.
[12] 孙华飞,崔玉衡.球面中具有平行第二基本形式的常平均曲率超曲面[J].辽宁大学学报(自然科学版),1992,19(2):12-17.
[13] 张运涛.Clifford环面(?)的刚性定理[J].数学物理学报(A),2008,28(1):128-132.
[14] Simons J.. Minimal varieties in Riemannian manifolds[J]. Ann. Math., 1968,88: 62-105.
[15] Yau S. T.. Submanifolds with constant mean curvature Ⅰ [J]. Amer.J.Math.,1974, 80: 346-366.
[16] 白正国,沈一兵.黎曼几何初步[M].高等教育出版社,2004.
[17] 陈维桓,李兴校.黎曼几何引论[M].北京大学出版社,2004.
[18] 赵正信.单位球面S~(n+1)(1)中平均曲率为非零常数的闭超曲面[J].淮北煤师院学报,2001,22(2):9-13.
[19] 袁斌贤.球面上具有常数量曲率的超曲面[J].中国科学技术大学学报,1998,28(6):696-700.
[20] 钟定兴,孙弘安.单位球面的超曲面的一个内蕴刚性定理[J].南昌大学学报理学版,2005,29(3):208-210.
[2] C. K. Peng, C. L. Terng. Minimal hypersurface of spheres with constant scalar curvature [J]. Ann. Math., 1983, 226: 177-198.
[3] C. K. Peng, C. L. Terng. The scalar curvature of minimal hypersurfaces in spheres [J]. Ann. Math., 1983, 226: 105-113.
[4] H. C. Yang, Q. M. Cheng. A note on the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere [J]. Chinise Science Bull, 1991, 36: 1-6.
[5] H. C. Yang, Q. M. Cheng. An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit spheres [J]. Manuscripta. Math., 1994, 82: 89-100.
[6] H. C. Yang, Q. M. Cheng. Chern's conjecture on minimal hypersurfaces [J].Math. Z., 1998, 227: 377-390.
[7] Q. M. Cheng. The rigidity of Clifford torus (?)[J]. Comment Math Helvetici. 1996, 71: 60-69.
[8] 魏嗣明,许洪伟.数量曲率的第二间隙[J].浙江大学学报(理学版),2007,34(4):375-377.
[9] 孙自琪.S~4内的常数量曲率的紧致超曲面[J].数学年刊(A),1987,8(3):273-286.
[10] 许洪伟.高维常平均曲率超曲面的数量曲率的间隙[J].高校应用数学学报(A),1993,8(4):410-419.
[11] 纪永强.子流形几何[M].北京:科学出版社,2004,208-211.
[12] 孙华飞,崔玉衡.球面中具有平行第二基本形式的常平均曲率超曲面[J].辽宁大学学报(自然科学版),1992,19(2):12-17.
[13] 张运涛.Clifford环面(?)的刚性定理[J].数学物理学报(A),2008,28(1):128-132.
[14] Simons J.. Minimal varieties in Riemannian manifolds[J]. Ann. Math., 1968,88: 62-105.
[15] Yau S. T.. Submanifolds with constant mean curvature Ⅰ [J]. Amer.J.Math.,1974, 80: 346-366.
[16] 白正国,沈一兵.黎曼几何初步[M].高等教育出版社,2004.
[17] 陈维桓,李兴校.黎曼几何引论[M].北京大学出版社,2004.
[18] 赵正信.单位球面S~(n+1)(1)中平均曲率为非零常数的闭超曲面[J].淮北煤师院学报,2001,22(2):9-13.
[19] 袁斌贤.球面上具有常数量曲率的超曲面[J].中国科学技术大学学报,1998,28(6):696-700.
[20] 钟定兴,孙弘安.单位球面的超曲面的一个内蕴刚性定理[J].南昌大学学报理学版,2005,29(3):208-210.