English

Geometric description of some Loewner chains with infinitely many slits

Complex Variables 2023-09-25 v1 Analysis of PDEs

Abstract

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers (bn)n1(b_n)_{n\ge1} and points of the real line (kn)n1(k_n)_{n\ge1}, we explicitily solve the Loewner PDE ft(z,t)=f(z,t)n=1+2bnzkn1t \dfrac{\partial f}{\partial t}(z,t)=-f'(z,t)\sum_{n=1}^{+\infty}\dfrac{2b_n}{z-k_n\sqrt{1-t}} in H×[0,1)\mathbb{H}\times[0,1). Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as t1t\rightarrow1^-.

Keywords

Cite

@article{arxiv.2309.12517,
  title  = {Geometric description of some Loewner chains with infinitely many slits},
  author = {Eleftherios Theodosiadis and Konstantinos Zarvalis},
  journal= {arXiv preprint arXiv:2309.12517},
  year   = {2023}
}