The multple-slit version of Loewner's differential equation and pointwise Hölder continuity of driving functions (Q2906160)

A bounded set \(A\subset\mathbb H=\{z\in\mathbb C: \text{Im}\;z>0\}\) is said to be a hull if \(A=\mathbb H\cap\overline A\) and \(\mathbb H\setminus A\) is simply connected. Let a family of growing hulls \(\{K_t\}_{t\in[0,E]}\) be parameterized so that there are conformal maps \(g_t:\mathbb H\setminus K_t\to\mathbb H\) with \(g_t(z)=z+{2t\over z}+O(|z|^{-2})\) near infinity. Solutions of the multiple slit version of Loewner's differential equation NEWLINE\[NEWLINE\dot g_t=\sum_{j=1}^n\frac{2\lambda_j(t)}{g_t(z)-U_j(t)},\;\;g_0(z)=z\in\mathbb H,NEWLINE\]NEWLINE with continuous functions \(\lambda_j,U_j: [0,E]\to\mathbb R\), \(\lambda_j(t)\geq0\), \(j=1,\dots,n\), \(\sum_{j=1}^n\lambda_j(t)=1\), represent conformal mappings \(g_t:\mathbb H\setminus K_t\to\mathbb H\).NEWLINENEWLINEDenote by \(\text{Lip}_L({1\over2})\) the set of all ``pointwise left \(1\over2\)-Hölder continuous'' functions \(U:[0,E]\to\mathbb R\), so that for every \(t\in[0,E]\), there are \(c>0\) and \(\epsilon>0\) such that \(|U(t)-U(s)|\leq c\sqrt{t-s}\) for all \(s\in[t-\epsilon,t]\). The main statement generalizes the sufficient slit condition for \(K_t\) which is known in the case \(n=1\).NEWLINENEWLINE Theorem 1.2: If \(U_j(t)<U_{j+1}(t)\) for all \(t\) and \(j=1,\dots,n-1\) and for every \(j\) and \(t\in(0,E]\), \(U_j\in\text{Lip}_L({1\over2})\) with NEWLINE\[NEWLINE\lim_{\epsilon\downarrow0}\sup\frac{|U_j(t)-U_j(t-\epsilon)|}{\sqrt\epsilon} <4\sqrt{\lambda_j(t)},NEWLINE\]NEWLINE then \(K_E\) consists of \(n\) disjoint simple curves.NEWLINENEWLINESay that the growing hulls \(K_t\) approach \(\mathbb R\) at \(U_j(0)\) in \(\varphi\)-direction, \(\varphi\in(0,\pi)\), if for every \(\epsilon>0\), there is a \(t_0>0\) such that the connected component of \(K_{t_0}\) having \(U_j(0)\) as a boundary point is contained in the set \(\{z\in\mathbb H: \varphi-\epsilon<\arg(z-U_j(0))<\varphi+\epsilon\}\). Then the author proves the following (Theorem 1.3):NEWLINENEWLINE Let \(j\in\{1,\dots,n\}\) and suppose \(U_j(0)\neq U_k(0)\) for all \(k\neq j\) and \(\lambda_j(0)\neq0\). The growing hulls \(K_t\) approach \(\mathbb R\) at \(U_j(0)\) in \(\varphi\)-direction if and only if NEWLINE\[NEWLINE\lim_{h\downarrow0}\frac{|U_j(h)-U_j(0)|}{\sqrt h}= \frac{2\sqrt{\lambda_j(0)}(\pi-2\varphi)}{\sqrt{\varphi(\pi-\varphi)}}.NEWLINE\]