The regularity of Loewner curves

Abstract

The Loewner differential equation, a classical tool that has attracted recent attention due to Schramm-Loewner evolution (SLE), provides a unique way of encoding a simple 2-dimensional curve into a continuous 1-dimensional driving function. In this thesis we study the curve in three cases according to the regularity of driving function: weakly H"older-1/2, H"older-1/2 with norm less than 4 and $C^alpha$ with $alphain (1/2,infty)$. In the first case, given the existence of the curve we show that the standard algorithm simulating the curve converges in a strong sense. One direct application is for simulating SLE. In the second case, we give another proof of Marshall, Rohde (2005) and Lind (2005) in which the curve exists and is a quasi-arc. A sufficient condition for the rectifiability of the curve is also given. In the final case, we show that the Loewner curve is in $C^{alpha+1/2}$. The thesis is a combination of three projects Tran (2013), Rohde-Zinsmeister-Tran (2013) and Lind-Tran (2014) which are joint work with Joan Lind, Steffen Rohde and Michel Zinsmeister.