On f-biharmonic maps and f-biharmonic submanifolds

f-Biharmonic maps are the extrema of the f-bienergy functional. f-biharmonic submanifolds are submanifolds whose defining isometric immersions are f-biharmonic maps. In this paper, we prove that an f-biharmonic map from a compact Riemannian manifold into a non-positively curved manifold with constant f-bienergy density is a harmonic map; any f-biharmonic function on a compact manifold is constant, and that the inversions about $S^m$ for $m\ge 3$ are proper f-biharmonic conformal diffeomorphisms. We derive f-biharmonic submanifolds equations and prove that a surface in a manifold $(N^n, h)$ is an f-biharmonic surface if and only it can be biharmonically conformally immersed into $(N^n,h)$. We also give a complete classification of f-biharmonic curves in 3-dimensional Euclidean space. Many examples of proper f-biharmonic maps and f-biharmonic surfaces and curves are given.