Parametric characteristics of high-order tangential Loewner's slits (Q1990018)

Denote the upper half-plane \(\{ z\in\mathbb{C}\colon\, \mathrm{Im}z>0\}\) by \(\mathbb{H}\) and let \(\Gamma\colon\, [0,T]\to \overline{\mathbb{H}}\) be a simple smooth curve such that \(\Gamma(0)=0\) and \(\Gamma(0,T]\subset\mathbb{H}\). For \(t\in[0,T]\), there exists a unique conformal map \(g(\cdot,t)\colon\, \mathbb{H}\setminus\Gamma(0,t]\to\mathbb{H}\) normalized hydrodynamically as \[ \lim\limits_{z\to\infty}(g(z,t) - z)\, =\, 0. \] Near \(\infty\), \(g(\cdot,t)\) has the expansion \(g(z,t)=z+b_1(t)z^{-1}+b_2(t)z^{-2}+\dots\) . Suppose that \(\Gamma\) is parameterized such that \(b_1(t)=2t\). Then \(g(\cdot,t)\) satisfies the Loewner differential equation in the upper half-plane \(\mathbb{H}\) \[ \frac{\partial g(z,t)}{\partial t}\, =\, \frac{2}{g(z,t)-\lambda(t)},\qquad g(z,0)\, =\, z, \] with a continuous real-valued function \(\lambda(t)=g(\Gamma(t),t)\) called the driving function. A class of driving functions \(\lambda\) is distinguished that, through the Loewner equation, generate analytic curves \(\Gamma\) which have \(n\)-order tangency to \(\mathbb{R}\) at \(0\), \(n\in\mathbb{N}\), \(n\geq 1\). An asymptotic expansion of \(\Gamma(t)\) as \(t\to 0+\) is obtained. A relation between harmonic measures of the two sides of \(\Gamma\) at \(g^{-1}(i,t)\) is found.