半黎曼卷积流形中的类空超曲面

发布时间：2017-12-28 10:38

本文关键词：半黎曼卷积流形中的类空超曲面　出处：《大连理工大学》2016年博士论文　论文类型：学位论文

【摘要】：由于半黎曼流形中类空超曲面在数学和物理方面的重要意义,一直被众多几何拓扑学家所关注.近年来,关于类空超曲面浸入到半黎曼卷积空间R×f Mn(ε=±1)中的唯一性的研究吸引了越来越多的学者的关注并取得了丰硕的成果.本文在众多学者的研究成果基础上,通过应用Omori-Yau极大法则和Stokes定理的推广来研究类空超曲面浸入到半黎曼卷积流形中的唯一性定理.拓展了卷积函数和高阶平均曲率的取值范围,得到如下主要研究结果：首先,对超曲面的高阶平均曲率和高度函数的梯度范数,我们分别给出合适的取值条件,在此条件下,得到了在广义的Robertson-Walker时空(后面均简记为GRW时空)上的刚性定理.其次,当外围空间为Lorentzian卷积流形-R×fMn时,本文对其上的卷积函数的导数f'为零和非零两种情况分别进行了讨论.当f'为零时,在其超曲面上应用推广的极大法则,研究当纤维Mn的截面曲率有下界时乘积流形-R×Mn的超曲面的唯一性；当f'非零时,应用极大法则和Stokes定理推论,得到了高阶平均曲率非零情况下超曲面的唯一性,并给出其在-R×tHn和-R×cosht Sn等空间上的应用.最后,当平均曲率非零时,应用经典的Omori-Yau极大法则得到了一定条件下黎曼卷积流形R×Mn上角度函数和卷积函数的导数之间的符号关系式,并推广至高阶平均曲率.应用此关系式和Stokes定理的推论,研究了黎曼卷积空间上高阶平均曲率和卷积函数的导数均非零的条件下整体图的唯一性.此外,还给出了此唯一性在(-π/2,-π/2)×cost Hn,和R×cosht Hn等空间上的应用.
[Abstract]:The significance of semi Riemann Submanifolds in space like Hypersurfaces in mathematics and physics, has been much concerned by numerous geometric topology. In recent years, a spacelike hypersurface immersed into semi Riemann space R * f Mn convolution (E = 1) in the study of uniqueness has attracted more and more scholars pay attention to and achieved fruitful results. Based on many scholars on the research results, through the application of Omori-Yau law greatly and Stokes theorem to study the spacelike Hypersurfaces in the uniqueness theorem of semi Riemann manifolds. Convolution expands the range of convolution function and higher mean curvature, the main results are as follows first, gradient norm of high-order mean curvature hypersurfaces and height functions, we give the appropriate condition, under this condition, obtained in generalized Robertson-Walker space (denoted as G were behind RW) the space-time rigidity theorem. Secondly, when the outer space for the Lorentzian convolution manifold -R * fMn, the derivative F'on the convolution function on the zero and 02 conditions are discussed respectively. When F' is zero, the great law application in its hypersurface, uniqueness of hypersurfaces the study of fiber Mn with sectional curvature when the lower product manifold -R * Mn; when F'is nonzero, inference using maximum principle and Stokes theorem, the higher mean curvature of non uniqueness in the case of zero hypersurface, and given its -R * tHn and -R * cosht Sn space application finally, when the average curvature is not zero, the Omori-Yau rule has been greatly applied classical symbol relation between derivative Riemann convolution manifold R * Mn angle function and convolution function under certain conditions, it is generalized to higher mean curvature. Application of this formula and Stokes theorem The corollary is that we study the uniqueness of global graphs under the condition that the derivatives of higher order mean curvature and the convolution function are all nonzero under Riemann's convolution space. In addition, we also give the application of this uniqueness in the spaces of (- PI /2, PI /2) x cost Hn, R * cosht Hn and so on.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O186.1

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