Biharmonic hypersurfaces with constant scalar curvature in space forms

TL;DR

This paper investigates biharmonic hypersurfaces with constant scalar curvature in space forms, showing they have constant mean curvature or are minimal under certain conditions, and confirms parts of Chen's conjecture.

Contribution

It proves that such hypersurfaces have constant mean curvature or are minimal depending on the ambient space curvature, partially confirming Chen's conjecture.

Findings

No proper biharmonic hypersurfaces with constant scalar curvature in Euclidean or hyperbolic space for dimensions less than 7.

Hypersurfaces in positive curvature space forms have constant mean curvature.

Hypersurfaces in non-positive curvature space forms are minimal.

Abstract

Let $M^n$ be a biharmonic hypersurface with constant scalar curvature in a space form $\mathbb M^{n+1}(c)$. We show that $M^n$ has constant mean curvature if $c>0$ and $M^n$ is minimal if $c\leq0$, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^{n+1}$ or hyperbolic space $\mathbb H^{n+1}$ for $n<7$.