Harmonic measures of slit sides perpendicular to the domain boundary

TL;DR

This paper investigates the geometric properties of solutions to the chordal Loewner equation, demonstrating that harmonic measures of perpendicular slit sides in the upper half-plane become asymptotically equal.

Contribution

It provides a novel analysis of harmonic measures for slit sides perpendicular to the boundary, advancing understanding of the Loewner equation's geometric solutions.

Findings

Harmonic measures of perpendicular slit sides are asymptotically equal.

The study compares singular solutions and harmonic measures in slit domains.

Results contribute to the geometric theory of Loewner evolution.

Abstract

The article is devoted to the geometry of solutions to the chordal Loewner equation which is based on comparison of singular solutions and harmonic measures for the sides of a slit in domains generated by a driving term. It is proved that harmonic measures of two sides of a slit in the upper half-plane which is perpendicular to the real axis are asymptotically equal to each other.