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

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

2003-07-04

arxiv.org/abs/math/0307066v1

J. Differential Geometry 72 (3): 467-508, March 2006

Mathematics

Differential Geometry

33 pages, 8 figures

Scientific paper

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 $\pi$. 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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