Exact solutions of Loewner equations (Q2211144)

The author considers an exact solution to the chordal Loewner differential equation and gives an expression of the corresponding driving function. For the upper half-plane \(\mathbb H=\{z\in\mathbb C:\text{Im}\,z>0\}\) and \(T>0\) let \(\gamma[0,T]\) be a simple curve, \(\gamma(0)\in\mathbb R\), \(\gamma(0,T]\subset\mathbb H\). There is a conformal map \(g_t\) from \(H_t=\mathbb H\setminus\gamma[0,t]\), \(0\leq t\leq T\), onto \(\mathbb H\) such that \[g_t(z)=z+\frac{2t}{z}+O\left(\frac{1}{|z|^2}\right),\;\;\;z\to\infty.\] In this case, \(g_t(z)\) satisfies the chordal Loewner differential equation \[\frac{\partial}{\partial t}g_t(z)=\frac{2}{g_t(z)-\lambda(t)},\;\;\;g_0(z)=z\in\mathbb H,\] where the driving function \(\lambda(t)=\lim_{z\to\gamma(t)}g_t(z)\) is continuous and real-valued. On the other hand, given a continuous driving function \(\lambda\), a solution \(g_t\) to the Loewner equation generates an increasing family of hulls. The author considers a sequence of slits \[\gamma^p:=\left\{(1+e^{i\theta})^p:\pi\left(1-\frac{1}{2p}\right)\leq\theta\leq\pi\right\},\;\;\;p=4n+1,\;\;\;n\in\mathbb N,\] which are the traces generated by some driving functions. The main result of the paper is given in the following theorem. Theorem 1.1. Let \(\gamma^p\) be the trace generated by the chordal Loewner differential equation. Then there is a constant \(C>0\) such that the driving function \(\lambda\) is of the form \[\lambda(t)=Ct^{\frac{p+1}{2p}}+o\left(t^{\frac{p+1}{2p}}\right),\;\;\;t\to0.\]