Geometric Analysis Of Ball - Type Quaternary Tangential Manifold

We investigate spherical qc manifolds by studying conformal qc geometry. We construct the qc Yamabe operators on qc manifolds, which are covariant under the conformal transformations. A qc manifold is scalar positive, negative or vanishing if and only if its qc Yamabe invariant is positive, negative or zero, respectively. On a scalar positive spherical qc manifold, we can construct the Green function of the qc Yamabe operator, which can be applied to construct a conformally invariant tensor. It becomes a spherical qc metric if the qc positive mass conjecture is true. Conformal geometry of spherical qc manifolds can be applied to study convex cocompact subgroups of Sp(n+1,1).In Chapter 1, we give a comprehensive survey of the backgrounds and mod-ern developments of contact manifolds, the Yamabe problem, convex cocompact subgroups, chains and R-circles. And then we introduce main ideas and concepts used in this thesis and our main results.In Chapter 2, we introduce the basic facts about qc manifolds, the quater-nionic Heisenberg group, the quaternionic hyperbolic space, Sp(n+1,1), spherical qc manifolds and connected sums.In Chapter 3, we construct the qc Yamabe operator and its Green function, and investigate their properties.In chapter 4, we use the Green function of the qc Yamabe operator to con-struct a conformally invariant tensor and propose the qc positive mass conjecture. We point out if the qc positive mass conjecture is true, it becomes a spherical qc metric. We prove that some connected sums of two scalar positive spherical qc manifolds are also scalar positive. So scalar positive spherical qc manifolds are abundant.In chapter 5, we recall the definitions of the convex cocompact subgroups of Sp(n+1,1) and the Patterson-Sullivan measure. Then we define an invariant metric on Ω(Γ)/Γ,where Ω(Γ)=S4n+3\Λ(Γ) and Λ(Γ) is the limit set of Γ. As a corollary, we prove that the spherical qc manifold Ω(Γ)/Γ is scalar positive, negative or vanishing if and only if the Poincare critical exponent of the discrete subgroup Γ is less than, greater than or equal to 2n+2, respectively.In chapter 6, we define the chain and R-circle on the quaternionic Heisen-berg group, and give the property of chains under the vertical projection. We also prove the uniqueness of the chain passing through two distinct points, qc-horizontality of R-circles and give the relationship between R-circle and pure imaginary R-circle.