Screw Motion Invariant Minimal Surfaces from Gluing Helicoids

Abstract

We consider families of embedded, screw motion invariant minimal surfaces in R 3 which limit to parking garage structures. We derive balance equations for the nodal limit and regenerate to obtain surfaces corresponding to solutions. We thus prove the existence of many new examples with helicoidal or planar ends.

Helicoids and vortices

“…Rotating vortex crystals correspond to screw-motion invariant minimal surfaces; this connection is readily established in [TW05,Fre21]. Our main results in Section 2 establish the other cases of the claimed connection.…”

“…The connection between rotating vortex crystals and screw-motion invariant minimal surfaces is established in the following theorem [Fre21].…”

“…Example 2 (Polygonal configuration). A Karcher-Scherk tower with 2n wings can be twisted into a configuration of n negatively handed helicoids lying at the vertices of a polygon; see Figure 1 and [Fre21,Prop. 8.11].…”

“…8.11]. The Fischer-Koch surfaces can be twisted into a similar configuration, only with an extra helicoid, positively or negatively handed, at the center of the polygon; see [Fre21,Prop. 8.13].…”

“…For simplicity, they assumed that the helicoids are aligned along a straight line, and noticed that the roots of Hermite polynomials provide examples of balanced and non-degenerate configurations. Recently, the second named author implemented a similar construction without this assumption [Fre21].…”