数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 981-993.doi: 10.1007/s10473-023-0301-6
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Xin Wei1,†, Zhi-Ying Wen2
Xin Wei1,†, Zhi-Ying Wen2
摘要: Let Γ be a Jordan curve in the complex plane and let Γλ be the constant distance boundary of Γ. Vellis and Wu \cite{VW} introduced the notion of a (ζ,r0)-chordal property which guarantees that, when λ is not too large, Γλ is a Jordan curve when ζ=1/2 and Γλ is a quasicircle when 0<ζ<1/2. We introduce the (ζ,r0,t)-chordal property, which generalizes the (ζ,r0)-chordal property, and we show that under the condition that Γ is (ζ,r0,√t)-chordal with 0<ζ<r1−√t0/2, there exists ε>0 such that Γλ is a t-quasicircle once Γλ is a Jordan curve when 0<ζ<ε. In the last part of this paper, we provide an example: Γ is a kind of Koch snowflake curve which does not have the (ζ,r0)-chordal property for any 0<ζ≤1/2, however Γλ is a Jordan curve when ζ is small enough. Meanwhile, Γ has the (ζ,r0,√t)-chordal property with 0<ζ<r1−√t0/2 for any t∈(0,1/4). As a corollary of our main theorem, Γλ is a t-quasicircle for all 0<t<1/4 when ζ is small enough. This means that our (ζ,r0,t)-chordal property is more general and applicable to more complicated curves.