Constant mean curvature Isometric Immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and related results

TL;DR

This paper classifies constant mean curvature isometric immersions with constant intrinsic curvature into product spaces like 2  R and related manifolds, revealing new examples and applications of surface correspondences.

Contribution

It provides a comprehensive classification of such immersions and surfaces in various 3D homogeneous manifolds, extending existing theories and introducing new examples.

Findings

Classification of isometric immersions with constant intrinsic curvature

New examples of constant mean curvature surfaces in product manifolds

Applications of surface correspondences to classify surfaces in () and () manifolds.

Abstract

In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the $3-$dimensional homogenous manifolds $\mathbb{E}(\kappa, \tau)$ and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \times \mathbb{H}^2$ with constant intrinsic curvature. It is worthwhile to point out that these classifications provide new examples.