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 \(\gamma(1)\in\mathbb H\).