Driving functions and traces of the Loewner equation

Abstract

We consider the chordal Loewner differential equation in the upper half-plane, the behavior of the driving function λ(t) and the generated hull K _t when K _t approaches λ(0) in a fixed direction or in a sector. In the case that the hull K _t is generated by a simple curve γ(t) with γ(0) = 0, we prove some sharp relations of ${{\lambda (t)} \mathord{\left/ {\vphantom {{\lambda (t)} {\sqrt t }}} \right. \kern-\nulldelimiterspace} {\sqrt t }}$ and ${{\gamma (t)} \mathord{\left/ {\vphantom {{\gamma (t)} {\sqrt t }}} \right. \kern-\nulldelimiterspace} {\sqrt t }}$ as t → 0 which improve the previous work.