A Loewner Equation for Infinitely Many Slits

Technau, Marc; Technau, Niclas

doi:10.1007/s40315-016-0179-6

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…A recent, detailed proof is given by A. Monaco and P. Gumenyuk in [15]. See also A. Starnes [24] and M.-N. Technau [25] for the multiple and infinitely many slits versions respectively. Assume that γ is a Jordan curve emanating from R, with parameterization γ " γptq, 0 ď t ă 8 and write K t :" γpr0, tsq.…”

Section: Introductionmentioning

confidence: 99%

Explicit multi-slit Loewner flows and their geometry

Theodosiadis

2023

Preprint

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In this paper we present explicit solutions to the radial and chordal Loewner PDE and we make an extensive study of their geometry. Specifically, we study multi-slit Loewner flows, driven by the time-dependent point masses $\mu_{t}:=\sum_{j=1}^{n}b_j \delta_{\{\zeta_j e^{iat}\}}$ in the radial case and $\nu_t:=\sum_{j=1}^{n}b_j\delta_{\{k_j\sqrt{1-t}\}}$ in the chordal case, where all the above parameters are chosen arbitrarily. Furthermore, we investigate their close connection to the semigroup theory of holomorphic functions, which also allows us to map the chordal case to the radial one.

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Section: Introductionmentioning

confidence: 99%

Explicit multi-slit Loewner flows and their geometry

Theodosiadis

2023

Preprint

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Geometric Description of Some Loewner Chains with Infinitely Many Slits

Theodosiadis,

Zarvalis

2024

J Geom Anal

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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 $$(b_n)_{n\ge 1}$$ ( b n ) n ≥ 1 and points of the real line $$(k_n)_{n\ge 1}$$ ( k n ) n ≥ 1 , we explicitily solve the Loewner PDE $$\begin{aligned} \dfrac{\partial f}{\partial t}(z,t)=-f'(z,t)\sum _{n=1}^{+\infty }\dfrac{2b_n}{z-k_n\sqrt{1-t}} \end{aligned}$$ ∂ f ∂ t ( z , t ) = - f ′ ( z , t ) ∑ n = 1 + ∞ 2 b n z - k n 1 - t in $$\mathbb {H}\times [0,1)$$ H × [ 0 , 1 ) . Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as $$t\rightarrow 1^-$$ t → 1 - .

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Explicit Multi-slit Loewner Flows and Their Geometry

Theodosiadis

2024

Comput. Methods Funct. Theory

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In this paper we present explicit solutions to the radial and chordal Loewner PDEs and we make an extensive study of their geometry. Specifically, we study multi-slit Loewner flows, driven by the time-dependent point masses $$\mu _{t}:=\sum _{j=1}^{n}b_j \delta _{\{\zeta _j e^{iat}\}}$$ μ t : = ∑ j = 1 n b j δ { ζ j e iat } in the radial case and $$\nu _t:=\sum _{j=1}^{n}b_j\delta _{\{k_j\sqrt{1-t}\}}$$ ν t : = ∑ j = 1 n b j δ { k j 1 - t } in the chordal case, where all the above parameters are chosen arbitrarily. Furthermore, we investigate their close connection to the semigroup theory of holomorphic functions, which also allows us to map the chordal case to the radial one.

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A Loewner Equation for Infinitely Many Slits

Cited by 3 publications

References 12 publications

Explicit multi-slit Loewner flows and their geometry

Explicit multi-slit Loewner flows and their geometry

Geometric Description of Some Loewner Chains with Infinitely Many Slits

Explicit Multi-slit Loewner Flows and Their Geometry

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