Screw motion invariant minimal surfaces from gluing helicoids

Abstract

The purpose of this paper is to construct one parameter families of embedded, screw motion invariant minimal surfaces in $${\mathbb {R}}^3$$ which limit to parking garage structures. We construct such surfaces by defining Weierstrass data on the quotient and closing the periods. In the nodal limit, the periods reduce to algebraic balance equations for the locations of the helicoidal nodes. For any configuration of nodes that solve the equations and satisfy a nondegeneracy condition, we regenerate to obtain a family of surfaces near the limit. We thus prove the existence of many new examples of surfaces near the nodal limit, with helicoidal or planar ends. Among these are candidates for genus g helicoids distinct from those currently known. We do not require any symmetry for the solutions of the balance equations, which suggests the existence of helicoidal surfaces only symmetric with respect to screw motions. This introduces new directions for the study and classification of screw motion invariant surfaces.