Driving functions and traces of the Loewner equation
Suppose that a simple curve \(\gamma:[0,T]\to\mathbb C\), \(\gamma(t)\in\mathbb H:=\{z:\Im z>0\}\), except for \(\gamma(0)=0\), is parameterized so that a conformal map \(g_t\) from \(\mathbb H_t:=\mathbb H\setminus\gamma[0,t]\) onto \(\mathbb H\) is expanded near infinity as \[ g_t(z)=z+\frac{2t}{z}+O\left(\frac{1}{|z|^2}\right),\qquad z\to\infty. \] In this case, \(g_t(z)\) satisfies the chordal Loewner differential equation \[ \dot g_t(z)=\frac{2}{g_t(z)-\lambda(t)},\qquad g_0(z)=z, \] with a continuous real-valued driving function \(\lambda(t)\). On the other hand, given a continuous function \(\lambda:[0,T]\to\mathbb R\) and \(z\in\mathbb H\), a solution \(g_t\) to the Loewner equation is the unique conformal transformation from \(\mathbb H_t:=\mathbb H\setminus K_t\) onto \(\mathbb H\), where \(\{K_t\}_{t\in[0,T]}\) is a family of growing hulls in \(\mathbb H\) generated by \(\lambda\). The authors investigate the relationship between the growing hulls \(K_t\) and the driving function \(\lambda(t)\) when \(t\to0\). Theorem 1.1. Let the Loewner equation generate a slit \(\gamma\), and let \(\lambda(t)/\sqrt t\to c\in\mathbb R\) as \(t\to0^+\). Then \[ \lim_{t\to0^+}\sup|\gamma(t)|/\sqrt t=|b(\theta)|, \] where \[ b(\theta):=2\left(\frac{\pi}{\theta}-1\right)^{\frac{1}{2}-\frac{\theta}{\pi}}e^{i\theta}\quad\text{and}\quad\theta:=\theta(c)=\frac{\pi}{2}\left(1-\frac{c}{\sqrt{c^2+16}}\right). \] Theorem 1.2. Let the Loewner equation generate a slit \(\gamma\), where \(\gamma(t)\) is continuously differentiable and \(\gamma'(t)\) is nonvanishing on \((0,T]\), and let \(\lim_{t\to 0^+}\arg\gamma'(t)=\theta\in(0,\pi)\). Then \[ \lim_{t\to0^+}\frac{\gamma(t)}{\sqrt t}=b(\theta),\quad\lim_{t\to0^+}\frac{s(t)}{\sqrt t}=|b(\theta)|\quad\text{and}\quad\lim_{t\to0^+} \frac{\lambda(t)}{\sqrt t}=c(\theta), \] where \(s(t)\) is the arc-length of \(\gamma[0,t]\) and \(c(\theta)\) is the inverse of \(\theta(c)\). Extend the consideration to a sector: \(K_t\) approaches 0 in a sector if there exist angles \(\alpha,\beta\in(0,\pi)\) and \(d>0\) such that \(\arg K_d:=\{\arg z: z\in K_d\}\subset(\alpha,\beta)\). Theorem 1.3. Let the growing hulls \(K_t\) be generated by the Loewner equation and let \(K_t\) approach 0 in a sector. Then \[ \lim_{t\to0^+}\sup\frac{|\lambda(t)|}{\sqrt t}<\infty. \]
scientific article; zbMATH DE number 6376157
scientific article