A circle homeomorphism is Weil-Petersson if and only if the Beltrami differential of its unique quasiconformal harmonic extension is square-integrable.
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Harmonic extension of Weil-Petersson circle homeomorphisms
A circle homeomorphism is Weil-Petersson if and only if the Beltrami differential of its unique quasiconformal harmonic extension is square-integrable.