The Harmonic Mapping Whose Hopf Differential Is a Constant

Mathematics

Liang Shen¹ 

[1] School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China;

关键词: harmonic mapping;    Hopf differential;    Beltrami coefficient;   

DOI  :  10.3390/math8081310

来源: DOAJ

【 摘 要 】

Suppose that $h\left(z\right)$ is a harmonic mapping from the unit disk $\mathbb{D}$ to itself with respect to the hyperbolic metric. If the Hopf differential of $h\left(z\right)$ is a constant $c>0$, the Beltrami coefficient $\mu \left(z\right)$ of $h\left(z\right)$ is radially symmetric and takes the maximum at $z=0$. Furthermore, the mapping $\gamma :c\to \mu \left(0\right)$ is increasing and gives a homeomorphism from $(0,+\infty )$ to $(0,1)$.

【 授权许可】

Unknown