Lets say we want to find the extremal length of the family of curves $\Gamma$, which we say is $\mathcal{L}(\Gamma)$. $\Gamma$ moves through two adjacent Riemann manifolds, which have different metrics. Lets say these manifolds are $S_1$ and $S_2$. Each of them has its own conformal metric: $$ |\mathrm{d}z_1|: S_1\rightarrow \mathbb{R}_{\geq 0}$$ $$ |\mathrm{d}z_1|: S_2\rightarrow \mathbb{R}_{\geq 0}$$ We can say that $\gamma\in\Gamma$, and that the subsets of curves through the two manifolds are $\gamma_1\in\Gamma_1$ and $\gamma_2\in\Gamma_2$.
We define one edge of manifold $S_1$ as source of $\Gamma$, and the interface between the manifolds as the (local)sink. For $S_2$ we use it the other way around. That way the global source is $S_1$ and the global sink is $S_2$.
As they have different metrics the local extremal lengths cannot be added together (except for simple geometries of both).
Is it possible to find the extremal length for such a configuration as a whole?
P.S. I'm an electrical engineer, and just starting with this area of math
EDIT: Recently I learned about:
Carathéodory, C. (1912) Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten. Math. Ann. 72: pp. 107-144
Link to the article on Springer(German).
It seems to describe how to combine two conformal maps, specifically talking about continuity on the boundary as special case (which is possible). According to it, a mapping between the boundaries can be created after the mappings of the regions is done.
I would really like if someone more knowledgeable could comment on this.
Not in terms of the extremal lengths of the parts. It depends on how they are put together. E.g., if the shorter curves in $S_1 $ match up with shorter curves in $S_2$, the extremal length of the whole thing is smaller than if you flipped $S_2$, matching shorter curves in $S_1$ with longer curves in $S_2$.
So you actually have to look at the whole thing. At which point your question becomes "how do we find extremal length, anyway?"... the answer being that we usually can't, except for very simple geometries (as you noted). One comes up with a lower estimate by constructing an admissible density $\rho $, and with an upper estimate by somehow Fubinizing the integral of $\rho^n$.