Biconservative hypersurfaces with constant scalar curvature in space forms

TL;DR

This paper classifies biconservative hypersurfaces with constant scalar curvature in space forms, showing they have constant mean curvature in four dimensions and are either rotational or constant mean curvature in five dimensions.

Contribution

It solves an open problem by classifying such hypersurfaces in space forms for dimensions up to five.

Findings

Hypersurfaces in $N^4(c)$ have constant mean curvature.

In $N^5(c)$, hypersurfaces are either rotational or have constant mean curvature.

Abstract

Biconservative hypersurfaces are hypersurfaces which have conservative stress-energy tensor with respect to the bienergy, containing all minimal and constant mean curvature hypersurfaces. The purpose of this paper is to study biconservative hypersurfaces $M^n$ with constant scalar curvature in a space form $N^{n+1}(c)$. We prove that every biconservative hypersurface with constant scalar curvature in $N^4(c)$ has constant mean curvature. Moreover, we prove that any biconservative hypersurface with constant scalar curvature in $N^5(c)$ is ether an open part of a certain rotational hypersurface or a constant mean curvature hypersurface. These solve an open problem proposed recently by D. Fetcu and C. Oniciuc for $n\leq4$.