Constructions of helicoidal minimal surfaces and minimal annuli in $\widetilde{E(2)}$

TL;DR

This paper constructs new families of minimal surfaces in the Lie group E9(2), including helicoids and catenoids, using a Weierstrass-type representation, and explores their limits to prove a half-space theorem.

Contribution

It introduces two one-parameter families of minimal surfaces in E9(2), expanding the understanding of minimal surface constructions in this Lie group.

Findings

Construction of helicoidal and catenoidal minimal surfaces in E9(2)

Analysis of the limit of catenoidal minimal surfaces

A new proof of the half-space theorem for E9(2)

Abstract

In this article, we construct two one-parameter families of properly embedded minimal surfaces in a three-dimensional Lie group $\widetilde{E(2)}$, which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrass-type representation introduced by Meeks, Mira, P\'erez and Ros for minimal surfaces in Lie groups of dimension three. In the end, we study the limit of the catenoidal minimal surfaces. As an application of this limit case, we get a new proof of a half-space theorem for minimal surfaces in $\widetilde{E(2)}$.