New results toward the classification of biharmonic submanifolds in 𝕊ⁿ

Abstract We prove some new rigidity results for proper biharmonic immer- sions in Sⁿ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fun- damental form; hypersurfaces satisfying intrinsic properties; PMC sub- manifolds; parallel submanifolds.