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

In [2], two of the authors and W. Meeks found examples of translation-invariant, embedded minimal surfaces with an infinite number of topological ends. For each k > 0, a surfaceMk was constructed, which was invariant with respect to a translation parallel to the x3-axis, and under a rotation group of order k+ 1 around the x3-axis. The method of construction was generalized in [3] to obtain the first known examples of embedded, singly-periodic minimal surfaces with an infinite number of topological ends, invariant under screw-motions (with a nontrivial rotational component). For each integer k > 0 and angle θ, with |θ| < π k+1 , there exists an embedded surface Mk,θ whose orientation-preserving symmetry group contains a rotation of order k+1 around the x3-axis and a screw motion—a unit translation in the x3-direction, followed by a 2θ rotation around it. See Figure 0. Although the surfaces Mk,θ were conceived Partially supported by the Rhodes Trust. Supported by research grant DEFG02-86ER25015 of the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, and National Science Foundation, Division of Mathematical Sciences research grants DMS-9011083 and DMS-9101903. Partially supported by Sonderforschungsbereich SFB256 at Bonn.