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Minimal disks bounded by three straight lines in Euclidean space and trinoids in hyperbolic space

Benoit Daniel
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Abstract

Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than $π$. In the case of general angles, we prove that there exist at most four such minimal disks, we give a sufficient condition of existence in terms of a system of three equations of degree 2, and we give explicit formulas for the Weierstrass data in terms of hypergeometric functions. Finally, we construct constant-mean-curvature-one trinoids in hyperbolic space by the method of the conjugate cousin immersion.

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Benoit Daniel. 2003-07-04. Minimal disks bounded by three straight lines in Euclidean space and trinoids in hyperbolic space. https://arxiv.org/abs/math/0307066

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