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

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 \(\mathbb{S}^n(c)\times\mathbb{R}\).