【作者】 黄翠；

【导师】 杨宗信；

【作者基本信息】 江西师范大学 ， 基础数学， 2015， 硕士

【摘要】 John区域的概念是F.John在研究平面弹性理论时引入的.由于John区域与区域的多种度量有着密切关系,因此John区域已经成为复动力系统、逼近论和弹性理论中的重要研究对象.2013年,Willy由定义在单位圆B内的调和映射的Weierstrass-Enneper提升曲面提出了John曲面的概念.平面调和映照与John曲面的研究和三维流形几何有直接关系,因此研究平面调和映射的Schwarz导数和满足一定条件的调和映射的Weierstrass-Enneper提升曲面与John曲面的关系,是几何函数论的重要课题.全文共分为四个部分:第一部分,预备知识.简要介绍了John区域和Schwarz导数等相关概念以及平面调和映照的发展历史与研究现状,并简单的介绍作者的主要工作.第二部分,根据Pokornyi关于解析函数的单叶性准则,结合二阶微分方程解的比较定理,讨论了当单位圆内的非解析调和函数的Schwarz导数、共形因子和高斯曲率满足一定条件时,这个调和函数的像域的提升曲面是John曲面.第三部分,对于单位圆内的非解析调和函数,我们定义了新的共形因子,得到了新的Schwarz导数和高斯曲率,讨论了调和函数的提升映射的单叶性条件并得到了提升映射的两点偏差定理.更多还原

【Abstract】 The concept of John domain was introduced by F.John when he studied the elastic theory in plane. John domain is an important object of study in complex dynamic system, approximation theory and elastic theory because of the close relationship in John domain and various measurements of the region. According to the image of the unit disk under the canonical lift of a harmonic mapping, Willy proposed the concept of John surface in 2013. Because harmonic mappings and John surfaces have much to do with the geometries of 3-manifolds, the study of the Schwarzian derivative of a planar harmonic mapping and John surfaces associated with the Weierstrass-Enneper lift of a class of harmonic mappings are important subjects of geometric function theory.There are three parts in this article.The first part is the preface. In this part, we introduce the concept of John domain as well the development and the research situation of harmonic mapping in the plane and its Schwarzian derivative. The main results of this article are briefly introduced in this chapter.In part 2, on the basis of Pokornyi’s univalence criteria, we prove the image of the unit disk under the canonical lift of a harmonic mapping to be a John surface when the Schwarzian derivative and the conformal factor of the harmonic mapping satisfy certain condition by using a comparison theorem of second order differential equation.In part 3, we introduce a definition of the conformal factor of a unanalytic harmonic mapping in the unit disc, and discuss the univalence property of lift of the harmonic mapping, and also obtain the two-point distortion theorem for the harmonic mapping.更多还原