ON AHLFORS' IMAGINARY SCHWARZIAN
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Date
2021
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Abstract
We study geometric aspects of the imaginary Schwarzian S2f for curves in 3-space, as introduced by Ahlfors in [1]. We show that S2f points in the direction from the center of the osculating sphere to the point of contact with the curve. We also establish an important law of transformation of S2f under Mobius transformations. We finally study questions of existence and uniqueness up to Mobius transformations of curves with given real and imaginary Schwarzians. We show that curves with the same generic imaginary Schwarzian are equal provided they agree to second order at one point, while prescribing in addition the real Schwarzian becomes an overdetermined problem.
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Ahlfors' Schwarzian for curves, imaginary Schwarzian, osculating sphere, Mobius transformation, overdetermined problem