Vertically invariant minimal surfaces in unimodular semidirect products

TL;DR

This paper studies vertically invariant minimal surfaces in unimodular semidirect product Lie groups, providing new examples especially in the group (2) and analyzing their geometric properties.

Contribution

It characterizes and constructs new examples of vertically invariant minimal surfaces in unimodular semidirect product Lie groups, expanding understanding of their geometry.

Findings

New examples of minimal surfaces in (2) are described.

Vertical invariance leads to specific minimal surface classifications.

The work enhances the understanding of minimal surfaces in unimodular Lie groups.

Abstract

A surface in a three-dimensional metric Lie group $G$ is said invariant if it is invariant with respect to a one-dimensional subgroup $\Gamma$ of the isometry group of $G$. Is this work we focus on unimodular metric Lie groups $G$ that can be written as a semidirect product of the form $\mathbb{R}^2\rtimes_A \mathbb{R}$ for certain matrix $A\in \mathcal{M}_2(\mathbb{R})$ and study the minimal surfaces which are invariant under the group $\Gamma$ generated by left translations by elements in the vertical axis $\{0\}\rtimes\mathbb{R}$. We will call these surfaces vertically invariant. In particular, we describe new examples of minimal surfaces in $\widetilde{E}(2)$ which are vertically invariant.