数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 981-993.doi: 10.1007/s10473-023-0301-6

• •    下一篇

CONSTANT DISTANCE BOUNDARIES OF THE $t$-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE*

Xin Wei1,†, Zhi-Ying Wen2   

  1. 1. School of Science, Xi'an Shiyou University, Xi'an 710065, China;
    2. Department of Mathematics, Tsinghua University, Beijing 100080, China
  • 收稿日期:2021-12-09 修回日期:2022-06-29 出版日期:2023-06-25 发布日期:2023-06-06
  • 通讯作者: Xin Wei, E-mail: xwei@xsyu.edu.cn
  • 作者简介:Zhi-Ying Wen, E-mail: wenzy@mail.tsinghua.edu.cn
  • 基金资助:
    Wen was supported by the NSFC (12071167).

CONSTANT DISTANCE BOUNDARIES OF THE $t$-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE*

Xin Wei1,†, Zhi-Ying Wen2   

  1. 1. School of Science, Xi'an Shiyou University, Xi'an 710065, China;
    2. Department of Mathematics, Tsinghua University, Beijing 100080, China
  • Received:2021-12-09 Revised:2022-06-29 Online:2023-06-25 Published:2023-06-06
  • Contact: Xin Wei, E-mail: xwei@xsyu.edu.cn
  • About author:Zhi-Ying Wen, E-mail: wenzy@mail.tsinghua.edu.cn
  • Supported by:
    Wen was supported by the NSFC (12071167).

摘要: Let $\Gamma$ be a Jordan curve in the complex plane and let $\Gamma_\lambda$ be the constant distance boundary of $\Gamma$. Vellis and Wu \cite{VW} introduced the notion of a $(\zeta,r_0)$-chordal property which guarantees that, when $\lambda$ is not too large, $\Gamma_\lambda$ is a Jordan curve when $\zeta=1/2$ and $\Gamma_\lambda$ is a quasicircle when $0<\zeta<1/2$. We introduce the $(\zeta,r_0,t)$-chordal property, which generalizes the $(\zeta,r_0)$-chordal property, and we show that under the condition that $\Gamma$ is $(\zeta,r_0,\sqrt t)$-chordal with $0<\zeta < r_0^{1-\sqrt t}/2$, there exists $\varepsilon>0$ such that $\Gamma_\lambda$ is a $t$-quasicircle once $\Gamma_\lambda$ is a Jordan curve when $0<\zeta<\varepsilon$. In the last part of this paper, we provide an example: $\Gamma$ is a kind of Koch snowflake curve which does not have the $(\zeta,r_0)$-chordal property for any $0<\zeta\le 1/2$, however $\Gamma_\lambda$ is a Jordan curve when $\zeta$ is small enough. Meanwhile, $\Gamma$ has the $(\zeta,r_0,\sqrt t)$-chordal property with $0<\zeta < r_0^{1- \sqrt t}/2$ for any $t\in (0,1/4)$. As a corollary of our main theorem, $\Gamma_\lambda$ is a $t$-quasicircle for all $0<t<1/4$ when $\zeta$ is small enough. This means that our $(\zeta,r_0,t)$-chordal property is more general and applicable to more complicated curves.

关键词: constant distance boundary, $t$-quasicircle, Koch snowflake curve

Abstract: Let $\Gamma$ be a Jordan curve in the complex plane and let $\Gamma_\lambda$ be the constant distance boundary of $\Gamma$. Vellis and Wu \cite{VW} introduced the notion of a $(\zeta,r_0)$-chordal property which guarantees that, when $\lambda$ is not too large, $\Gamma_\lambda$ is a Jordan curve when $\zeta=1/2$ and $\Gamma_\lambda$ is a quasicircle when $0<\zeta<1/2$. We introduce the $(\zeta,r_0,t)$-chordal property, which generalizes the $(\zeta,r_0)$-chordal property, and we show that under the condition that $\Gamma$ is $(\zeta,r_0,\sqrt t)$-chordal with $0<\zeta < r_0^{1-\sqrt t}/2$, there exists $\varepsilon>0$ such that $\Gamma_\lambda$ is a $t$-quasicircle once $\Gamma_\lambda$ is a Jordan curve when $0<\zeta<\varepsilon$. In the last part of this paper, we provide an example: $\Gamma$ is a kind of Koch snowflake curve which does not have the $(\zeta,r_0)$-chordal property for any $0<\zeta\le 1/2$, however $\Gamma_\lambda$ is a Jordan curve when $\zeta$ is small enough. Meanwhile, $\Gamma$ has the $(\zeta,r_0,\sqrt t)$-chordal property with $0<\zeta < r_0^{1- \sqrt t}/2$ for any $t\in (0,1/4)$. As a corollary of our main theorem, $\Gamma_\lambda$ is a $t$-quasicircle for all $0<t<1/4$ when $\zeta$ is small enough. This means that our $(\zeta,r_0,t)$-chordal property is more general and applicable to more complicated curves.

Key words: constant distance boundary, $t$-quasicircle, Koch snowflake curve