关于调和映射的研究

The main aim of this project is to study: (1) The extreme points and support points of harmonic Bloch mappings: Because of the closely connection between Bloch functions and Bloch constants, Seidel and Walsh investigated (analytic) Bloch functions. After that, Cima, Wogen, Bonk, etc. discussed the existence of extreme points and support points of Bloch spaces. It is known that a harmonic mapping is Bloch if and only if both the analytic part and the anti-analytic part of the harmonic mapping are Bloch functions. We plan to use this property to study the existence of extreme points and support points of harmonic Bloch mappings, and therefore generalize the corresponding known results; (2) Uniformly local univalence of harmonic mappings and differential subordination of p-harmonic mappings: We plan to discuss the properties of uniformly locally univalent harmonic mappings; We have defined p-harmonic mappings in the plane. We will make the further researches on these mappings, and then establish relatively complete theory of p-harmonic mappings. This research has the important theoretical significance.

本项目主要研究以下两个方面的内容。（一）调和Bloch映射的极值点和支点：基于Bloch函数和Bloch常数的紧密联系，Seidel和Walsh开始了（解析）Bloch函数的研究。随后，Cima、Wogen、Bonk等讨论了Bloch空间的极值点、支点的存在性。众所周知，调和映射是Bloch映射当且仅当其解析部分和反解析部分都是Bloch函数，我们将利用此结论来研究调和Bloch映射极值点、支点的存在性等性质，从而推广已有相关讨论。（二）调和映射的一致局部单叶性和p-调和映射的微分从属：主要研究一致局部单叶调和映射的一些相关性质；我们已经定义了p-调和映射，此项目将对此作进一步研究，从而建立较完善的p-调和映射理论。此研究具有重要的理论意义。

项目：“关于调和映射的研究(No. 11226092)”主要讨论了调和映射、双调和映射以及多调和映射的性质，在研究工作中取得了较好的成绩。主要工作如下：(一) n-维复空间上的多调和映射：把调和映射的相关结果推广到多调和映射的情形，通过建立新的方法给出了Lipschitz型空间中的多调和映射的几个特征。(二)单位圆盘上单叶保向的双调和映射：作为星形双调和映射和凸双调和映射的推广，给出了一个单叶保向的双调和映射类，利用系数不等式给出了双调和映射属于此类的一个充分条件，且进一步证得此系数不等式是此类中具有非负系数的子类的特征。