The Normality Of Harmonic Mappings

The normality is one of main research objects of the geometric function theory,which includes the normality of a family of functions or the normality of one function.After the Nevanlinna theory bulit a connection between the normal family of meromorphic functions and the value of derivatives of functions,the development of this theory was rapidly promoted.The normality of mappings has been widely used in many fields such as complex dynamical systems,quasi-conformal mappings,minimal surfaces theory and so on.First,as a kind of generalization to holomorphic functions,the theory of harmonic mappings is one of main research objects in complex analysis.In this paper,we will continue to study the problem of normality for a family of harmonic functions.We utilize the generalized Hurwitz theorem and Marty criterion to give the Zalcman-Pang lemma for the family of harmonic functions,and then extend the Zalcman-Pang principle to the case of the family of harmonic functions.Then,as a generalization of Euclidean harmonic mappings in the weighted sense,the theoretical research of (?)-harmonic mapping has also been concerned recently.In this paper,we study the normality of (?)-harmonic mapping.By an expression of (?)-harmonic mappings,we use Bloch function,pre-Schwarz derivative and M-linearly connected domain to generalize some results about normality from the case of Euclidean harmonic mappings to the one of (?)-harmonic mappings.Finally,similar to meromorphic functions,there are also many value distribution problems such as Bloch heuristic principle in the surface theory.In this paper,the value distribution of Gauss mappings of Weingarten surfaces with finite total curvature is also studied.We utilize the generalized Riemann-Hurwitz relation and Jorge-Meeks formula to obtain a Picard type theorem for Gauss mappings of ESWMT-surfaces.