A Family of Singly Periodic Minimal Surfaces Invariant under a Screw Motion

Michael Callahan David Hoffman Hermann Karcher

Abstract

We construct explicitly, using the generalized Weierstrass representation, a complete embedded minimal surface M_{k,θ} invariant under a rotation of order k + 1 and a screw motion of angle 2θ about the same axis, where k > 0 is any integer and θ is any angle with |θ| < π/(k+1). The existence of such surfaces was proved in [Callahan et al. 1990], but no practical procedure for constructing them was given there.

We also show that the same problem for θ = ±π/(k + 1) does not have a solution enjoying reflective symmetry; the question of the existence of a solution without such symmetry is left open.