Collisions and spirals of Loewner traces. (English) Zbl 1206.30024

Let \(\mathbb H\) be the upper half-plane, and let \(g_t:\mathbb H\setminus K_t\to\mathbb H\) be the solution to the Loewner equation
\[ \frac{dg_t(z)}{dt}=\frac{2}{g_t(z)-\lambda(t)},\qquad g_0(z)=z, \]
with a continuous real-valued driving term \(\lambda(t)\). The authors analyse the Loewner traces \(\gamma\) driven by \(\lambda\) asymptotically to \(k\sqrt{1-t}\).
Theorem 5.10: If a sufficiently smooth (e.g., asymptotically conformal) Loewner trace \(\gamma[0,1]\) has a self-intersection of angle \(\pi(1-\theta)\) with \(\theta\in[0,1)\), then
\[ \lim_{t\to1}\frac{\lambda(t)-\lambda(1)}{\sqrt{1-t}}=k, \]
where \(k=2\sqrt{1-\theta}+2/\sqrt{1-\theta}.\) If \(\gamma\) is asymptotically similar to the logarithmic spiral \(S_{\theta}(t)=\exp(te^{i\theta})\), then this limit holds with \(k=-4\sin\theta\).
For a compact set \(A\subset\mathbb H\) such that \(\big(\mathbb C\cup\{\infty\}\big)\setminus A\) is simply connected, let \(f\) be a conformal map of the unit disk \(\mathbb D\) onto \(\big(\mathbb C\cup\{\infty\}\big)\setminus A\) with \(f(0)=\infty\). Consider the curve \(v_0\) in \(\mathbb D\) given by \(v_0(t)=te^{i/(t-1)}\), \(0<t<1\). For a suitable \(t_0\), \(f\big(v_0(t)\big)\), \(t_0<t<1\), parameterizes a curve that begins in \(\mathbb R\) and winds around \(A\). Scale this curve considering the curve \(cf\circ v_0\) with an appropriate \(c\). Call the resulting curve \(v^A(t)\). A driving term \(\mu:[0,1)\to\mathbb R\) has local \(\text{Lip\,}1/2\) norm not less than \(C\) if there exists \(\delta>0\) such that \(|\mu(t)-\mu(t')|\leq C|t-t'|^{1/2}\) for all \(0\leq t<t'<1\) with \(|t-t'|<\delta(1-t)\).
Proposition 5.9: For \(v=v^A\), the driving term \(\lambda=\lambda^A\) has arbitrarily small local \(\text{Lip\,}1/2\) norm.
The important part of the paper is the proof of a form of stability of the self-intersection for \(\lambda(t)=k\sqrt{1-t}\).
Theorem 6.1: Suppose that
\[ \lim\limits_{t\to1}\frac{\lambda(t)}{\sqrt{1-t}}=k>4, \] and assume that there exists \(C<4\) such that \(\lambda\) has local \(\text{Lip\,}1/2\) norm less than \(C\). Then the trace \(\gamma[0,1]\) driven by \(\lambda\) is a Jordan arc. Moreover, \(\gamma_T(1)\in\mathbb R\) and
\[ \lim_{t\to1}\arg\big(\gamma_T(t)-\gamma_T(1)\big)=\pi\frac{1-\sqrt{1-16/k^2}}{1+\sqrt{1-16/k^2}}, \]
provided \(1-T\) is sufficiently small.
Here \(\gamma_T\) equals \(g_T\big(\gamma[T,1]\big)\big/\sqrt{1-T}\). The method of the proof of Theorem 6.1 also applies to the case \(|k|<4\). In this case, the trace \(\gamma\) driven by \(\lambda\) is a Jordan arc, and \(\gamma\) is asymptotically similar to the logarithmic spiral at \(\gamma(1)\in\mathbb H\).


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C20 Conformal mappings of special domains
30C62 Quasiconformal mappings in the complex plane
30C30 Schwarz-Christoffel-type mappings
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