Simons' Type Equation in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$ and Applications

TL;DR

This paper derives Simons' type equations for constant mean curvature surfaces in product spaces like ^2  and ^2 , characterizes certain immersions, and explores their geometric properties.

Contribution

It introduces new Simons' type equations for specific immersions in product spaces and characterizes particular classes of these immersions.

Findings

Derived Simons' type equations for ^2  and ^2  spaces.

Characterized specific immersions with constant mean curvature.

Provided applications to geometric analysis of these surfaces.

Abstract

Equations of Simons type are presented. They are satisfied by a pair of special operators associated to the immersion $\Sigma^2\looparrowright M^2(c)\times\mathbb{R}$ with constant mean curvature. Some immersions are characterized.