Let $\gamma_0$, $\gamma_1$ be disjoint Jordan arcs on $\overline{\mathbb{C}}$, where $\gamma_0$ has endpoints at $a_1$ and $a_2$ and $\gamma_1$ has endpoints at $a_3$ and $a_4$. Two configurations of arcs $\{\gamma_0,\gamma_1\}$ and $\{\gamma'_0,\gamma'_1\}$ are called equivalent if $\gamma_k$ and $\gamma'_k$, $k=0,1$, have the same end points and if the arcs $\gamma_0$, $\gamma_1$ can be deformed to the arcs $\gamma'_0$, $\gamma'_1$ by the isotopy of $\overline{\mathbb{C}}$ that keeps the endpoints of the arcs fixed. For fixed $a_1$, $a_2$, $a_3$, $a_4$, there are countable number of equivalence classes of configurations $\{\gamma_0,\gamma_1\}$, which we denote as $\Gamma_j$, $j=1,2,\ldots$ A configuration is called canonical if for each $k\in \{0,1\}$, the arc $\gamma_k$ is a hyperbolic geodesic in the simply connected domain $\overline{\mathbb{C}}\setminus \gamma_{1-k}$. The following result was recently discussed by M. Bonk and A. Eremenko (Canonical embeddings of pairs of arcs. arXiv:2101.00088v1 [math.CV] 31 Dec 2020.)

Theorem. Every equivalence class $\Gamma_j$ contains a unique canonical configuration.

In this talk, we show how this result follows from Jenkins's theorem on extremal partitioning of Riemann surfaces. Few related problems, including O. Teichmüller's problem on the minimal capacity of condensers with fields separating given pairs of points and L. Ahlfors's problem on separation of two continua by circles, also will be discussed.

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