从属理论在平面调和映射中的应用研究

In recent years, planar harmonic mapping has became a hot research topic in the cross field of complex analysis and geometry. It has a close connection with quasiconformal mapping, differential geometry and geometric partial differential equation, and it is widely used in image processing and statistical physics. This research project aims to study the following contents. Firstly, by virtue of subordination theory and shear method, we shall derive the univalency of convex combinations of planar harmonic mappings and try to solve the problem posed by Dorff. Secondly, we will discuss maximum valency problems of planar harmonic mappings by means of weak subordination theory, approximation theory and argument principle. Thirdly, we shall establish theories of boundary subordination and weak boundary subordination of planar harmonic mappings with the aid of Julia function, and discuss boundary properties of planar harmonic mappings by using these theories. By proposing and solving the problems in the project, it will help to deepen studies of the subordination theory, and it will extend applications of this theory in related fields.

近年来，平面调和映射已成为复分析与几何交叉领域中的热门研究课题。它与拟共形映射、微分几何及几何偏微分方程有着紧密联系，同时在图像处理和统计物理学等方面也有着广泛应用。本项目拟研究以下内容：1) 融合从属理论和剪切方法，研究平面调和映射凸组合的单叶性，着重解决Dorff所提出的公开问题；2) 综合运用弱从属理论、逼近理论及幅角原理，探讨平面调和映射中的最大叶数问题；3) 根据Julia函数建立平面调和映射的边界从属和弱边界从属理论，利用这些理论研究平面调和映射的边界性质。本项目中问题的提出和解决，将有助于深化从属理论本身的研究并拓展该理论在相关领域中的应用。

调和映射理论与拟共形映射、微分几何及几何偏微分方程有着紧密的联系，是现代数学的重要研究方向之一。本项目主要研究了以下三个方面的内容：一是研究了平面调和映射的凸组合的单叶性问题，得到了平面调和映射的若干近于凸判别准则，并部分的解决了M. Dorff所提出的一个问题；二是讨论了平面调和映射的最大叶数问题，得到了满足一定条件的某类平面调和映射的近于凸半径，部分的解决了该类映射的最大叶数问题；三是利用从属理论和边界从属理论，得到了平面调和映射的若干边界性质，并得到了某类近于凸调和映射的系数估计、增长定理、覆盖定理及面积定理。相关结果对平面调和映射的从属问题、单叶性问题、最大叶数问题及边界问题的研究具有较重要的作用。