Space-filling curves and phases of the Loewner equation (Q2874077)

Let \(\lambda(t)\) be a real-valued continuous function on \([0,T]\). The chordal Loewner equation NEWLINE\[NEWLINE \frac{\partial g_t(z)}{\partial t}=\frac{2}{g_t(z)-\lambda(t)},\;\;g_0(z)=z, NEWLINE\]NEWLINE NEWLINE\[NEWLINE g_t(z)=z+\frac{2t}{z}+O\left(\frac{1}{z^2}\right),\;\;z\to\infty, NEWLINE\]NEWLINE has a solution on a time interval for each \(z\in\mathbb H:=\{z:\mathrm{Im} z>0\}\). The solution \(g_t(z)\) is extended continuously onto \(\mathbb H\cup(\mathbb R\setminus\lambda(0))\) and maps \(\mathbb H\setminus K_t\) onto \(\mathbb H\), where \(K_t\) are the compact sets called the hulls generated by the driving term \(\lambda\). There is a close analogy between the behavior of the Schramm-Loewner evolution traces and the hulls \(K_t\) driven by functions \(\lambda\in\text{Lip}({1\over2})\). In the article, the authors prove the existence of a phase transition for the Loewner equation.NEWLINENEWLINETheorem 1.1. Suppose \(\lambda\) is a \(\text{Lip}({1\over2})\) driving function that generates a curve with nonempty interior. Then \(\|\lambda\|_{{1\over2}}\geq4.0001\). If the requirement of the trace being space filling is given up, then, for every sequence \(z_1,z_2,\dots\) of points in \(\mathbb H\), there is a trace \(\gamma\) that visits these points and a driving function has \(\text{Lip}({1\over2})\) norm at most 4.NEWLINENEWLINE The authors provide a rather general criterion for hulls to be driven by \(\text{Lip}({1\over2})\) functions and obtain a large class of examples of such hulls.NEWLINENEWLINETheorem 1.3. Let \(\{K_t\}\) be a family of hulls generated by the driving term \(\lambda(t)\) for \(t\in[0,T]\). Suppose there is some \(C_0>0\) and \(k<\infty\) so that for each \(s\in[0,T)\), there exists a \(k\)-quasi-disk \(D_s\subset\mathbb H\) with \(\infty\in\overline D_s\) such that the following properties hold:NEWLINENEWLINE(1) \(K_s\subset\mathbb H\setminus D_s\),NEWLINENEWLINE(2) \(K_T\setminus K_s\subset\overline D_s\),NEWLINENEWLINE(3) \(\text{diam}(K_t\setminus K_s)\leq C_0\max\{\text{dist}(z,\partial D_s)\;|\;z\in K_t\setminus K_s\}\) for all \(t\in(s,T]\).NEWLINENEWLINEThen \(\lambda\) is in \(\text{Lip}({1\over2})\) and \(\|\lambda\|_{{1\over2}}\leq C(k,C_0)\).NEWLINENEWLINECorollary 1.4. The van Koch curve, the half-Sierpinski gasket, and the Hilbert space-filling curve are all generated by \(\text{Lip}({1\over2})\) driving terms. There is a \(\text{Lip}({1\over2})\) driving term whose trace is a simple curve \(\gamma\) with positive area. In particular, this \(\gamma\) is not conformally removable, and therefore not uniquely determined by its conformal welding.