$f$-Biharmonic hypersurfaces into a conformally flat space

TL;DR

This paper investigates the properties and classifications of $f$-biharmonic hypersurfaces and surfaces in conformally flat and nonpositively curved spaces, revealing conditions for minimality and existence of proper $f$-biharmonic submanifolds.

Contribution

It provides new classifications and constructions of $f$-biharmonic hypersurfaces in conformally flat and negatively curved spaces, including conditions for minimality and proper $f$-biharmonicity.

Findings

Proper $f$-biharmonic hypersurfaces in nonpositively curved manifolds are noncompact.

Totally umbilical $f$-biharmonic surfaces in 3-manifolds with nonpositive sectional curvature are minimal.

Existence of proper $f$-biharmonic submanifolds of dimension $m eq4$ in negatively curved manifolds.

Abstract

We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We also explore $f$-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Secondly, we construct $f$-biharmonic surfaces and biharmonic conformal immersions of the associated surfaces into a conformall flat 3-space and also give a complete classification of $f$-biharmonic surfaces of nonzero constant mean curvature in 3-space forms. Finally, we especially investigate $f$-biharmonicity of hypersurfaces into a conformally flat space of negative sectional curvature. We show that any totally umbilical $f$-biharmonic surface of a 3-manifold with nonpositve sectional curvature is minimal whilst there are proper $f$-biharmonic $m$-dimensional submanifolds with $m\geq3$ and $m\neq4$ into nonpositvely curved manifolds.