【作者】 张东梅；

【导师】 周振荣；

【作者基本信息】 华中师范大学 ， 几何分析， 2009， 硕士

【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.更多还原

【Abstract】 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.更多还原