Universal uniqueness criterion?

[BenStandeven]

I guess its about deforming one initial region into the other initial region

How do we go about doing that? It is not given whether the Abel functions are holomorphic outside of their domains.

If you go slightly to the right of \( \partial_1 D_1 \) having a curve \( \gamma \) inside \( D_1 \) then you can continue \( A_1 \) to the region between \( \partial_2 D_1 \) and \( F(\gamma) \), i.e. \( A_1(F(z)):=A_1(z)+1 \) where we use the already established values of \( A_1 \) on the region between \( \partial_1 D_1 \) and \( \gamma \).

This works if \( F \) is bijective in the considered area. For example \( F=\exp \) is bijective in the strip along the real axis \( \{z: -\pi < \Im(z) < \pi \} \), which includes the fixed points \( L \) and \( L^\ast \).

Of course, this procedure might not help if the derivatives at the fixed points are positive reals (or are 0). In that case, near the fixed points, the expanded \( A_1 \) won't be any closer to \( A_2 \) than the original one was.