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Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be $$ \sup_{g \in \omega} \frac{\ell_g(\gamma)^2}{A_g(\Sigma)} $$ where $\ell_g(\gamma)$ is the (minimum) length of $\gamma$ in the metric $g$, $A_g(\Sigma)$ is the area of $\Sigma$ with respect to $g$, and the supremum runs over all metrics for which the area is finite. This is, of course, very well-studied. The definition extends without problem to cases when $\gamma$ is a homotopy class of graphs instead; that is, you are given an auxiliary graph $\Gamma$, and a homotopy class $\gamma$ of embeddings of $\Gamma$ in $\Sigma$. A natural generalization is to allow weighted graphs, where the edges of $\Gamma$ come with a weight $w(e)$. This weight is used in the definition of the length: for $C \in \gamma$, $$ \ell_g(C) = \sum_{e\text{ edge of }\Gamma} w(e) \ell_g(e). $$ (You then take the infimum over elements in the homotopy class, as usual.)

You can also extend this to homotopy classes relative to portions of the boundary.

Is it possible to give concrete metrics that realize the extremal length, as in the classical case of curves? For instance, in his book Conformal Invariants, Ahlfors essentially answers the question for tripod graphs embedded in the disk. That is, take the unit disk, and divide its boundary into three arcs. Let $\gamma$ be the class of tripods embedded in the disk with one leg in each of the three arcs. Then the metric realizing the extremal length with respect to this class turns out to be an equilateral triangle.

Have other examples been worked out? For instance, Maxime Fortier-Bourque and I worked out the case of weighted tripods, which turn out to be Euclidean triangles of other shapes (always acute or right triangles).

Here's one concrete question: suppose you look at a disk with the boundary divided into four segments, and look at embeddings of an "H" graph with one end in each of the segments. What's the metric that realizes the extremal length? There is one parameter in this problem (the modulus of 4 points on the boundary of a disk). For the symmetric case (realized, eg, by a square), the answer is a rhombus with angles 60-120-60-120. But what is the answer in general?

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