Hermite polynomials and helicoidal minimal surfaces

Abstract

The main objective of this paper is to construct smooth 1-parameter families of embedded minimal surfaces in euclidean space that are invariant under a screw motion and are asymptotic to the helicoid. Some of these families are significant because they generalize the screw motion invariant helicoid with handles and thus suggest a pathway to the construction of higher genus helicoids. As a byproduct, we are able to relate limits of minimal surface families to the zero-sets of Hermite polynomials.