Asymptotic ratio of harmonic measures of slit sides

TL;DR

This paper investigates the asymptotic behavior of harmonic measures on the sides of slits in the upper half-plane generated by the chordal Loewner equation, revealing specific ratios for tangential slits.

Contribution

It introduces new asymptotic ratios of harmonic measures for slits with particular tangency properties, advancing understanding of slit geometry in complex analysis.

Findings

Asymptotic ratio for tangential slits to a straight line at a given angle

Asymptotic ratio for slits with high order tangency to circular arcs

Enhanced understanding of harmonic measure behavior in slit geometry

Abstract

The article is devoted to the geometry of solutions to the chordal Loewner equation which is based on the comparison of singular solutions and harmonic measures for the sides of a slit in the upper half-plane generated by a driving term. An asymptotic ratio for harmonic measures of slit sides is found for a slit which is tangential to a straight line under a given angle, and for a slit with high order tangency to a circular arc tangential to the real axis.