Helicoidal minimal surfaces of prescribed genus

TL;DR

This paper constructs and analyzes minimal surfaces of arbitrary genus in the product space S^2 x R, showing their convergence to helicoidal minimal surfaces in R^3 as the sphere's radius increases.

Contribution

It demonstrates the existence of genus-g minimal surfaces in S^2 x R with specified asymptotic helicoidal ends and their convergence to helicoidal minimal surfaces in R^3.

Findings

Existence of genus-g minimal surfaces in S^2 x R with prescribed helicoidal ends

Convergence of these surfaces to helicoidal minimal surfaces in R^3 as sphere radius grows

Helicoidal minimal surfaces of every genus appear as limits in the product space

Abstract

For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in $R^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in $R^3$ of every prescribed genus occur as such limits of examples in $S^2\times R$.