Biconservative hypersurfaces with constant scalar curvature in space forms

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 \(n\le 4\).