球型四元切触流形的几何分析

发布时间：2018-01-05 09:46

本文关键词：球型四元切触流形的几何分析　出处：《浙江大学》2016年博士论文　论文类型：学位论文

【摘要】：我们通过研究共形几何来研究球型四元切触(spherical qc)流形.我们构造了四元切触流形上的四元Yamabe算子,它在共形变换下是协变的.一个四元切触流形称为数量曲率为正,负,零的,当且仅当它的Yamabe不变量是正,负,零的.在数量曲率为正的球型四元切触流形上,我们可以构造四元切触Yamabe算子的Green函数,并用它来构造一个共形不变量.如果四元切触正质量猜测成立的话,这是一个球型四元切触度量.球型四元切触流形上的共形几何可以用来研究Sp(n+1,1)的凸余紧子群.第一章中,我们介绍了四元切触流形,Yamabe问题,凸余紧子群以及四元Heisenberg群上的链和R-圆的历史背景和研究现状,同时介绍了本文的研究思想和主要结论.第二章中,我们介绍了四元切触流形,四元Heisenberg群,四元双曲空间,Sp(n+1,1)作用以及球型四元切触流形和连通和的基本概念及相关性质.第三章中,我们构造了四元切触Yamabe算子及其Green函数,并给出了相关性质.第四章中,我们用四元切触Yamabe算子的Green函数构造了一个共形不变张量并提出了四元切触正质量猜测.我们证明了如果四元切触正质量猜测成立的话,这是一个球型四元切触度量.同时我们还证明了两个数量曲率为正的球型四元切触流形的连通和的数量曲率也是正的.第五章中,我们回顾了Patterson-Sullivan测度的定义.对于Sp(n+1,1)的凸余紧子群Γ,我们构造了Q(Γ)/Γ上的不变度量.这里Ω(Γ)=S4n+3\Λ(Γ)且人(Γ)是Γ的极限集.我们证明了Q(Γ)/Γ的数量曲率是正,负,零的,当且仅当Γ的Poincare指数是大于,小于,等于2n+2.第六章中,我们定义了四元Heisenberg群上的链和R-圆,并给出了链在垂直投影下的性质.我们还证明了经过四元Heisenberg群上固定两点的链的唯一性,R-球的qc-水平性,并给出了R-圆与纯虚R-圆之间的关系.
[Abstract]:We study spherical Quaternary contact spherical QC manifolds by studying conformal geometry. We construct quaternion Yamabe operators on quaternion contact manifolds. A quaternionic contact manifold is called a quaternion whose curvature is positive, negative, zero if and only if its Yamabe invariant is positive and negative. On the spherical quaternion contact manifold with positive scalar curvature, we can construct the Green function of the quaternion contact Yamabe operator. It is used to construct a conformal invariant. If the quaternionic contact positive mass conjecture holds, this is a spherical quaternion contact metric. Conformal geometry on the spherical quaternionic contact manifold can be used to study Sp(n 1. In chapter 1, we introduce the problem of quaternion contact manifold called Yamabe. The historical background and research status of chain and R- circle on convex cocompact subgroups and quaternion Heisenberg groups are introduced. We introduce quaternion contact manifold, quaternion Heisenberg group and quaternion hyperbolic space. In chapter 3, we construct quaternion Yamabe operator and its Green function. The related properties are given in Chapter 4th. We construct a conformal invariant Zhang Liang by using the Green function of the quaternion contact Yamabe operator and propose a quaternion contact positive mass conjecture. We prove that if the quaternion contact positive quality conjecture holds. This is a spherical quaternion contact metric. At the same time, we prove that the connected sum of two spherical quaternion contact manifolds with positive scalar curvature is also positive in Chapter 5th. We review the definition of Patterson-Sullivan measure for the convex cocompact subgroup 螕. In this paper, we construct an invariant metric on Q (螕 ~ n / 螕), where 惟 (螕 ~ -S _ 4n _ 3\ A (螕)) is the limit set of 螕. We prove that the scalar curvature of Q (螕 _ n / 螕) is positive and negative. Zero, if and only if the Poincare exponent of 螕 is greater than, less than, equal to 2n 2. 6th, we define chains and R- circles on quaternion Heisenberg groups. We also prove the uniqueness of the chain with fixed two points over a quaternion Heisenberg group. The relation between R-circle and pure virtual R-circle is given.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O186.12

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