A large class inside the collection of properly embedded minimal surfaces in 3-dimensional Euclidean space \(\mathbb{R}^3\) is the one consisting of those surfaces which are invariant by an infinite cyclic group of direct rigid motions \(G\) acting freely on \(\mathbb{R}^3\), called single-periodic minimal surfaces. Such a surface can be viewed inside the flat three-manifold \(\mathbb{R}^3/G\). In [Indiana Univ. Math. J. 45, 177-204 (1996; Zbl 0864.53008)] J. Pérez and A. Ros introduced a nondegeneracy notion for a properly embedded minimal surface with finite total curvature in \(\mathbb{R}^3\) in terms of the space of Jacobi functions on the surface which have logarithmic growth at the ends, and proved that the set of nondegenerate minimal surfaces with fixed topology is a finite-dimensional real-analytic manifold.
In this paper, the spaces of nondegenerate properly embedded minimal surfaces in quotients of \(\mathbb{R}^3\) by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite-dimensional real-analytic manifolds. Riemann’s minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. The author also gives natural immersions of those spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.