Tangential Loewner hulls

TL;DR

This paper investigates the behavior of Loewner hulls generated by driving functions that approach zero at a specific rate, demonstrating their tangential behavior and trace existence for certain parameters.

Contribution

It provides a detailed analysis of the tangential behavior of Loewner hulls driven by functions approaching zero at a polynomial rate, extending understanding of their geometric properties.

Findings

Loewner hulls exhibit tangential behavior at time T for driving functions approaching zero as (T-t)^r with r in (0, 1/2)

Proves the existence of traces for these Loewner hulls under specified conditions

Demonstrates that the trace curves are tangential in the studied cases

Abstract

Through the Loewner equation, real-valued driving functions generate sets called Loewner hulls. We analyze driving functions that approach 0 at least as fast as $a (T-t)^r$ as $t \to T$, where $r \in (0, 1/2)$, and show that the corresponding Loewner hulls have tangential behavior at time $T$. We also prove a result about trace existence and apply it to show that the Loewner hulls driven by $a(T-t)^r$ for $r \in (0,1/2)$ have a tangential trace curve.