On biharmonic hypersurfaces with constant scalar curvatures in $\mathbb E^5(c)$

TL;DR

This paper proves that proper biharmonic hypersurfaces with constant scalar curvature in the 5-dimensional sphere must have constant mean curvature, and shows nonexistence in Euclidean and hyperbolic spaces, supporting Chen's conjecture.

Contribution

It establishes new conditions for biharmonic hypersurfaces in various constant curvature spaces, confirming parts of Chen's conjecture.

Findings

Proper biharmonic hypersurfaces in $ extbf{S}^5$ with constant scalar curvature have constant mean curvature.

No proper biharmonic hypersurfaces with constant scalar curvature exist in $ extbf{E}^5$ or $ extbf{H}^5$.

Results provide partial affirmation of Chen's and Generalized Chen's conjectures.

Abstract

We prove that proper biharmonic hypersurfaces with constant scalar curvature in Euclidean sphere $\mathbb S^5$ must have constant mean curvature. Moreover, we also show that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^5$ or hyperbolic space $\mathbb H^5$, which give affirmative partial answers to Chen's conjecture and Generalized Chen's conjecture.