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E. K. Theodosiadis: Geometric description of some Loewner chains with infinitely many slits

Time: Wed 2023-11-08 11.00 - 12.00

Location: Albano, Cramérrummet

Participating: E. K. Theodosiadis (SU)

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The chordal Loewner PDE describes the dynamics of a continuous and decreasing family of simply connected domains of the upper half-plane \(\mathbb{H}\). In this talk, we present and describe geometrically explicit solutions to the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers \((b_n)_{n \ge 1}\)and points of the real line \((k_n)_{n \ge 1}\), we explicitily solve the Loewner PDE

\(\displaystyle \frac{\partial f}{\partial t}(z, t) = - f'(z, t) \sum_{n = 1}^{+ \infty} \frac{2b_n}{z - k_n \sqrt{1-t}}\)

in \(\mathbb{H} \times [0,1)\), with initial value \(f(z,0) = z\). We also see that there is a close connection to the theory of semigroups of holomorphic self-maps of \(\mathbb{H}\), and using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as \(t \to 1^-\).