Remarks on the regularity of quasislits

TL;DR

This paper provides quantitative estimates on the geometric regularity of quasislits generated by Loewner equations, relating their cone containment, opening angles, and regularity to the Lip-1/2 seminorm of the driving function.

Contribution

It offers sharp bounds on the cone angles and regularity of quasislits based on the Loewner driving function's Lip-1/2 seminorm, advancing understanding of their geometric properties.

Findings

Sharp estimates on cone opening angles for s < 4

Explicit Hölder exponents for quasislits with s < 4

Quantitative relations between Lip-1/2 seminorm and quasiconformal dilatation

Abstract

A quasislit is the image of a vertical line segment [0, iy], y > 0, under a quasiconformal homeomorphism of the upper half-plane fixing infinity. Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-1/2 class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-1/2 seminorm, s. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the full range s <4. The estimate is shown to be sharp for small s. As consequences, we derive explicit H\"older exponents for s < 4 as well as estimates on winding rates. We also relate quantitatively the Lip-1/2 seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.