We construct a (non-removable) Jordan curve and a non-Mbius homeomorphism of the Riemann sphere which is conformal on the complement of and maps the curve onto itself. The curve is flexible in the sense of Bishop and may be taken to have zero area. The existence of such curves and conformal homeomorphisms is closely related to the non-injectivity of conformal welding.
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Malik Younsi (2018) studied this question.