Isoparametric surfaces in $\mathbb{E}(\kappa,\tau)$-spaces

TL;DR

This paper classifies specific types of surfaces in homogeneous 3-manifolds with 4-dimensional isometry groups, focusing on isoparametric, constant principal curvature, homogeneous, and constant mean curvature surfaces with special properties.

Contribution

It provides an explicit classification of four families of surfaces in these 3-manifolds, extending understanding of their geometric properties.

Findings

Classification of isoparametric surfaces in $ ext{E}( extkappa, au)$-spaces

Characterization of surfaces with constant principal curvatures

Analysis of surfaces with constant mean curvature and vanishing Abresch-Rosenberg differential

Abstract

We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and surfaces with constant mean curvature and vanishing Abresch-Rosenberg differential.