New results toward the classification of Biharmonic submanifolds in $\mathbb{S}^{n}$

TL;DR

This paper establishes new rigidity results for proper biharmonic submanifolds in spheres, including Dupin hypersurfaces, those with bounded second fundamental form, and submanifolds with specific intrinsic properties, advancing classification efforts.

Contribution

It provides novel rigidity theorems for various classes of proper biharmonic submanifolds in spheres, expanding the understanding of their geometric structure.

Findings

Rigidity results for Dupin hypersurfaces

Results for hypersurfaces with bounded second fundamental form

Classification insights for PMC and parallel submanifolds

Abstract

We prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.