Surfaces with parallel mean curvature in $\mathbb{S}^3\times\mathbb{R}$ and $\mathbb{H}^3\times\mathbb{R}$

TL;DR

This paper derives a Simons type equation for non-minimal surfaces with parallel mean curvature in certain 3D space forms times a line, and uses it to characterize specific complete pmc surfaces.

Contribution

It introduces a new Simons type equation for pmc surfaces in $M^3(c)\times\mathbb{R}$ and applies it to classify certain complete non-minimal pmc surfaces.

Findings

Derived a Simons type equation for pmc surfaces in $M^3(c)\times\mathbb{R}$

Characterized specific complete non-minimal pmc surfaces using the equation

Provided new insights into the geometry of pmc surfaces in product spaces

Abstract

We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.