On mapping problems in several complex variables

The thesis consists of two parts. In the first part, we study a regularity problem for CR mappings between CR manifolds. More precisely, we establish various versions of the Schwarz reflection principle in several complex variables. In particular, as a consequence of the main results, we confirm a conjecture of X. Huang in [Hu2] and provide a solution to a question raised by Forstneric [Fr1] (See Corollaries 2.1.11 and 2.1.12). It is a joint work with Shiferaw Berhanu ([BX1], [BX2]). In the second part, we study the embeddability problem from compact real algebraic strongly pseudoconvex hypersurfaces into a sphere. In a joint work with Xiaojun Huang and Xiaoshan Li ([HLX]), we prove that for any integer N, there is a family of compact real algebraic strongly pseudoconvex hypersurfaces in C2, none of which can be locally holomorphically embedded into the unit sphere in CN. This shows that the Whitney (or Remmert, respectively) type embedding theorem in differential topology (or in the Stein space theory, respectively) does not hold in the setting above.