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      Geometric description of some Loewner chains with infinitely many slits

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          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 \((b_n)_{n\ge1}\) and points of the real line \((k_n)_{n\ge1}\), we explicitily solve the Loewner PDE \[ \dfrac{\partial f}{\partial t}(z,t)=-f'(z,t)\sum_{n=1}^{+\infty}\dfrac{2b_n}{z-k_n\sqrt{1-t}}\] in \(\mathbb{H}\times[0,1)\). Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as \(t\rightarrow1^-\).

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          Journal
          21 September 2023
          Article
          2309.12517
          7919c44a-9e94-4228-8b1f-3a334f99ed44

          http://creativecommons.org/licenses/by/4.0/

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