Biharmonic submanifolds with parallel mean curvature in $\mathbb{S}^n\times\mathbb{R}$

TL;DR

This paper develops a Simons type formula for pmc submanifolds in product spaces and uses it to classify proper-biharmonic pmc surfaces in spheres times real lines, providing new insights into their geometric properties.

Contribution

It introduces a novel Simons type formula for pmc submanifolds in product spaces and classifies proper-biharmonic pmc surfaces in these settings.

Findings

Established a gap theorem for the mean curvature of certain biharmonic submanifolds.

Classified proper-biharmonic pmc surfaces in $ ext{S}^n(c) imes ext{R}$.

Derived new formulas linking curvature and biharmonicity in product spaces.

Abstract

We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^n(c)\times\mathbb{R}$.