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With the help of the Schwarz-Christoffel map, for a given polygon (given angle), we can find some points on the boundary of the upper half plane, such that a particular Schwarz-Christoffel map takes the upper half plane to the interior of given polygon conformally.

Can we ask similar questions: Like let the $(AB)$, $(CD)$ be intervals of $\mathbb R$ (boundary of the upper half plane), can we get a conformal map from the upper half plane to some domain in $\mathbb C\cup\infty$ such that $(AB)$ maps to some part the unit circle, $(CD)$ maps to some other part of the unit circle, but $(BC)$ doesn't map to the unit circle.

(basically, rather than fixing the angle (as in the Schwarz-Christoffel map), can we fix the modulus and free the angle)

Can we have some method to construct such a map explicitly?

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Take a curvilinear quadrilateral $A',B',C',D'$ satisfying all your conditions, that is $[A',B'],[C',D']$ are disjoint arcs of the unit circle, while the other sides do not belong to the unit circle. You can make the modulus of this quadrilateral arbitrary, by fixing $[A',B']$ and changing the length and position of $[C',D']$. So make it the same as the modulus of your quadrilateral $[A,B,C,D]$. Then there will be a conformal map $[A,B,C,D]$ onto $[A',B',C',D'].$

To obtain an almost explicit expression, choose the arcs which are not on the unit circle to be arcs of circles perpendicular to the unit circle. Then you have a circular quadrilateral, and the mapping function is a ratio of two solutions of the Heun (or Lame) differential equation, which is integrable (so the mapping function is expressed in terms of elliptic integrals). This is not completely explicit since you have to determine the accessory parameter. This can be done numerically.

You may look in MR3628295 Eremenko, Alexandre; Gabrielov, Andrei On metrics of curvature 1 with four conic singularities on tori and on the sphere. Illinois J. Math. 59 (2015), no. 4, 925–947.

where such conformal maps are studied.

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  • $\begingroup$ Thank you very much. I will go through the article. $\endgroup$
    – zapkm
    Commented Oct 20, 2022 at 16:46

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