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

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, give a sufficient condition of existence in terms of a system of three equations of degree 2, and 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.