The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Abstract

Given an integer k ≥ 2 k\geq 2 , let S ( k ) {\mathcal S}(k) be the space of complete embedded singly periodic minimal surfaces in R 3 \mathbb {R}^3 , which in the quotient have genus zero and 2 k 2k Scherk-type ends. Surfaces in S ( k ) {\mathcal S}(k) can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that S ( 2 ) {\mathcal S}(2) consists of the 1 1 -parameter family of singly periodic Scherk minimal surfaces. We prove that for each k ≥ 3 k\geq 3 , there exists a natural one-to-one correspondence between S ( k ) {\mathcal S}(k) and the space of convex unitary nonspecial polygons through the map which assigns to each M ∈ S ( k ) M\in {\mathcal S}(k) the polygon whose edges are the flux vectors at the ends of M M (a special polygon is a parallelogram with two sides of length 1 1 and two sides of length k − 1 k-1 ). As consequence, S ( k ) {\mathcal S}(k) reduces to the saddle towers constructed by Karcher.