For any twisted ideal polygon in H³, we construct a harmonic map from C to H³ with a polynomial Hopf differential, that is asymptotic to the given polygon, and is a bounded distance from a pleated plane. Our proof uses the harmonic map heat flow. We also show that such a harmonic map is unique once we prescribe the principal part of its Hopf differential.
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Gupta et al. (2024) studied this question.
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