本文關(guān)鍵詞：球型四元切觸流形的幾何分析　出處：《浙江大學(xué)》2016年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 四元切觸流形 球型四元切觸幾何 四元Green函數(shù) 四元Yamabe算子 凸余緊子群 鏈 R-球

【摘要】：我們通過研究共形幾何來研究球型四元切觸(spherical qc)流形.我們構(gòu)造了四元切觸流形上的四元Yamabe算子,它在共形變換下是協(xié)變的.一個(gè)四元切觸流形稱為數(shù)量曲率為正,負(fù),零的,當(dāng)且僅當(dāng)它的Yamabe不變量是正,負(fù),零的.在數(shù)量曲率為正的球型四元切觸流形上,我們可以構(gòu)造四元切觸Yamabe算子的Green函數(shù),并用它來構(gòu)造一個(gè)共形不變量.如果四元切觸正質(zhì)量猜測(cè)成立的話,這是一個(gè)球型四元切觸度量.球型四元切觸流形上的共形幾何可以用來研究Sp(n+1,1)的凸余緊子群.第一章中,我們介紹了四元切觸流形,Yamabe問題,凸余緊子群以及四元Heisenberg群上的鏈和R-圓的歷史背景和研究現(xiàn)狀,同時(shí)介紹了本文的研究思想和主要結(jié)論.第二章中,我們介紹了四元切觸流形,四元Heisenberg群,四元雙曲空間,Sp(n+1,1)作用以及球型四元切觸流形和連通和的基本概念及相關(guān)性質(zhì).第三章中,我們構(gòu)造了四元切觸Yamabe算子及其Green函數(shù),并給出了相關(guān)性質(zhì).第四章中,我們用四元切觸Yamabe算子的Green函數(shù)構(gòu)造了一個(gè)共形不變張量并提出了四元切觸正質(zhì)量猜測(cè).我們證明了如果四元切觸正質(zhì)量猜測(cè)成立的話,這是一個(gè)球型四元切觸度量.同時(shí)我們還證明了兩個(gè)數(shù)量曲率為正的球型四元切觸流形的連通和的數(shù)量曲率也是正的.第五章中,我們回顧了Patterson-Sullivan測(cè)度的定義.對(duì)于Sp(n+1,1)的凸余緊子群Γ,我們構(gòu)造了Q(Γ)/Γ上的不變度量.這里Ω(Γ)=S4n+3\Λ(Γ)且人(Γ)是Γ的極限集.我們證明了Q(Γ)/Γ的數(shù)量曲率是正,負(fù),零的,當(dāng)且僅當(dāng)Γ的Poincare指數(shù)是大于,小于,等于2n+2.第六章中,我們定義了四元Heisenberg群上的鏈和R-圓,并給出了鏈在垂直投影下的性質(zhì).我們還證明了經(jīng)過四元Heisenberg群上固定兩點(diǎn)的鏈的唯一性,R-球的qc-水平性,并給出了R-圓與純虛R-圓之間的關(guān)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.
【學(xué)位授予單位】：浙江大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O186.12

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本文編號(hào)：1382577