The driving function of the Loewner equation generating slit, emanating from a zero corner
Maher Karmushi, Ivan A. Kolesnikov, Yulia A. Loboda · Vestnik Tomskogo gosudarstvennogo universiteta Matematika i mekhanika · 2025
We construct a family of mappings ƒ = ƒ (z, τ) , τ∈[0, τ0]. When τ is fixed, the mapping ƒ translates the half-plane into a strip with a cut (the length of the cut depends on the parameter τ) along the ray γ going to infinity. The cut forms zero angles with the strip boundary. The decomposition of the governing function λ(τ) of the Loewner equation at the point τ = 0, τ > 0 generating such a family of regions is obtained. We formulate a hypothesis about the behavior of the control function generating a cut emerging from the zero corner of some single-connected region along the arc of a circle. The hypothesis is tested on one particular case.