Quasiconformal Mappings And CR Mappings On Real Submanifolds In C~n

This thesis focuses on the quasiconformal mappings on Carnot groups or Carnot-Caratheodory spaces,and the locally holomorphic automorphisms of real submanifolds in Cⁿ.We mainly discuss the rigidity of 1-quasiconformal mappings, and the explicit expression of the real-analytic infinitesimal CR automorphisms and the locally holomorphic automorphisms of real submanifolds in Cⁿ.In Chapter 1,we give a comprehensive survey of the backgrounds and modern developments of the Louville's theorem,quasiconformal mappings and the holomorphic automorphisms of real submanifolds.And then we introduce some concepts referring to this thesis and the main results of the subject.In Chapter 2,we mainly discuss the 1-quasiconformal mapping on two classes of Carnot groups:A(2,2)-type quadric Q₀(which is identical with a step two Carnot group) and the Engel group G.By the Beltrami equations satisfied by quasiconformal mappings,we show that the 1-quasiconformal mappings on Q₀ and G are CR or anti-CR.Furthermore,we get the unit component of the group of 1-quasiconformal mappings on Q₀ and the group of 1-quasiconformal mappings between the Engel group G,respectively.Chapter 3 is devoted to the study of quasiconformal mappings on a class of Carnot-Caratheodory spaces-strongly pseudoconvex smooth hypersurfaces in C^{n+ 1}.By approximating the CR structure at a point of such hypersurface by that of a local Heisenberg group,we show that the orientation preserving(or reserving) quasiconformal mappings on strongly pseudoconvex hypersurfaces satisfy a system of Beltrami equations.In particular,the 1-quasiconformal mappings on such surfaces are CR or anti-CR almost everywhere.Furthermore,if these surfaces are also real-analytic and non-spherical,then smooth 1-quasiconformal mappings on them,which fix a point,can be linearized.In the last chaper,by solving equations in the class of power series,we determines the real-analytic infinitesimal CR automorphisms of a class of nonhomogeneous rigid hypersurfaces F₁ in C^{N+ 1} and a class of pseudoconvex hy- persurfaces F₂ in C³ near the origin,respectively.And we get the connected component containing the identity transformation of all locally holomorphic automorphisms of F₁ near the origin and the unit connected component of the stability group at the origin of F₂.