f-Biharmonic maps and f-biharmonic submanifolds II

TL;DR

This paper advances the theory of f-biharmonic maps and submanifolds by characterizing harmonic maps, exploring conformality, and classifying f-biharmonic hypersurfaces and conformal immersions in various Riemannian manifolds.

Contribution

It introduces an improved equation for f-biharmonic hypersurfaces, establishes invariance under conformal changes, and provides classifications and rigidity results in specific geometric contexts.

Findings

Characterized harmonic maps via f-biharmonic maps.

Proved invariance of f-biharmonic maps under conformal metric changes.

Classified f-biharmonic hypersurfaces in Einstein spaces and space forms.

Abstract

We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian manifolds. We are able to characterize harmonic maps and minimal submanifolds by using the concept of f-biharmonic maps and prove that the set of all f-biharmonic maps from 2-dimensional domain is invariant under the conformal change of the metric on the domain. We give an improved equation for f-biharmonic hypersurfaces and use it to prove some rigidity theorems about f-biharmonic hypersurfaces in nonpositively curved manifolds, and to give some classifications of f-biharmonic hypersurfaces in Einstein spaces and in space forms. Finally, we also use the improved f-biharmonic hypersurface equation to obtain an improved equation and some classifications of biharmonic conformal immersions of surfaces into a 3-manifold.