Classification and Existence of f-Biharmonic Hypersurfaces in Nonpositively Curved Conformally Flat Spaces

We first study the f-biharmonicity of totally umbilical hypersurfaces in a Riemannian manifold of dimension n≥3 and prove that any totally umbilical proper f-biharmonic hypersurface without boundaries in a nonpositively curved manifold must be noncompact. Since biharmonic submanifolds are special cases of f-biharmonic submanifolds, our results on f-biharmonic hypersurfaces in nonpositively curved conformally flat spaces provide a natural extension of the generalized Chen’s conjecture. We then investigate the f-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Next, we study f-biharmonic surfaces in a conformally flat 3-space, and for those with nonzero constant mean curvature (CMC), we provide a complete classification of them in 3-space forms. Finally, we investigate the f-biharmonicity of hypersurfaces in a conformally flat space with negative sectional curvature. Our results generalize some previous conclusions on biharmonic hypersurfaces.