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.

The classical theory of minimal surfaces

“…As said before, the method also works for surfaces of infinite total curvature. For instance, Traizet and Weber [201] have used these ideas to perturb weak limits of planes and helicoids, thereby producing the first non-classical examples of minimal parking garage surfaces (see the comments after Theorems 4.5 and 4.7; also see the survey [116] for a more detailed explanation of this concept). Very recently, Traizet [200] has applied this method to produce a properly embedded minimal surface in R 3 with infinite genus and one limit end; see Footnote 43 for the importance of this new example.…”

The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

“…They also prove that if M ∈ M has finite positive genus and just one end, then it is asymptotic to the end of the helicoid and can be defined analytically in terms of meromorphic data on its conformal completion, which is a closed Riemann surface. This theoretical result together with the recent theorems developed by Weber and Traizet [39] and by Hoffman, Weber and Wolf [19] provide a theory on which a proof of Conjecture 2 might be based.…”

“…We refer the reader to section 11 of [26] where the basic theory of parking garage structures is developed, classical examples are given. We also refer the reader to [39] where it is shown that certain limiting minimal parking garage structures can be analytically untwisted via the implicit function theorem to produce one-parameter families of interesting periodic minimal surfaces that converge to it. …”

Properly embedded minimal surfaces with finite topology

“…Meeks and Rosenberg [38] have proposed the following question (which they proved for k = 0). There is a large number of examples constructed by desingularization, see for instance Kapouleas [21] and Traizet [61] for the non-periodic case and Traizet and Weber [62,64] for the periodic one.…”