Study On Geometric Theories Of Proper Mapping Of Complex Functions

Proper mapping is an important object in several complex variables,It is origi-nated from Stein and Remmert’s study of proper mappings on the complex space in the 1950s and 1960s. At present, it has become a rapidly growing popular branch of mathematics. The proper mapping of one variable is mainly considered in the Rie-mann surface. The study of the boundary behaviour of conformal mapping mainly concentrates in the 1970s and 1980s. In recent years, the rapid development of frac-tal geometry provides new tools, new methods, new ideas for the study of boundary behaviour of analytic functions. With the proper mappings appear, many mathe-maticians pay attention to the interection of the proper mapping, fractal geometry and boundary property which is the one of the hotspots of domestic and foreign scholars, and they had gained some achievements.In one complex variable function areas, we can easily know the Riemann map-ping is a special proper mapping whose degree is one. The role of Riemann exis-tence theorem is well-known in the one complex variable function areas. However, the existence of proper mapping of between multi-connectivity regions has not been resolved satisfactorily. Therefore, the promotion of Riemann existence theorem to the proper mapping is of great significance.In this thesis,we mainly study the existence problem of of proper mappings between two multi-connectivity regions. The existence of proper mapping between the general multi-connected region is very complicated, so we consider the special area in this paper, of course, there are also general conclusions (For example, the case of two multi-connectivity regions).In the first chapte of this thesis, we present the background of the study, definition of proper mapping, and related conclusions, also introduce the main conclusions obtained by us. The second chapter is to study the existence problem of holomorphic proper mapping for connectivity number≤2 of the domains, and the results obtained are perfect. The third chapter study the existence problem of holomorphic proper mapping for connectivity number≥2 of the domains,we solved this problem for a large class of domains.In the end of this thesis, we have given a valuation of coverage radius that an analytic function is a proper mapping in the neighborhood of the critical points, and the result is a generalization of Bloch theory.