2014 # On*f*-biharmonic maps and*f*-biharmonic submanifolds

**Abstract:** Abstractf -Biharmonic maps are the extrema of the f -bienergy functional. f -biharmonic submanifolds are submanifolds whose defining isometric immersions are fbiharmonic 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 ≥ 3 are proper f -biharmonic conformal diffeomorphisms. We de…

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“…To obtain examples of free f -biminimal surfaces, similar to Theorem 2.3 in [15], we state the following theorem:…”

confidence: 99%

“…To obtain examples of free f -biminimal surfaces, similar to Theorem 2.3 in [15], we state the following theorem:…”

confidence: 99%

“…) is a free f -biminimal surface. is f -biharmonic, then using Theorem 3.2 of [15] we get λ = 0 . Then the function f is indefinite, so this surface can not be f -biharmonic and free f -biminimal.…”

mentioning

confidence: 94%

“…A submanifold is called a f-biharmonic submanifold if its isometric immersion ϕ is f-biharmonic (cf. [8]). When f is a constant, f-biharmonic submanifolds are called biharmonic submanifolds (i.e., its bitension field τ 2 (ϕ) vanishes identically) (cf.…”

confidence: 99%

“…Naturally, the next step has been the study of f-biharmonic curves. Ou in [8] derived equations for f-biharmonic curves in a generic manifold and completely classified f-biharmonic curves in 3-dimensional Euclidean space E 3 , where he proved that such curves in E 3 are planar curves or general helices and gave some examples of nonbiharmonic f-biharmonic curves in E 3 . After that, there are a few valuable results on f-biharmonic curves in (generalized) Sasakian space forms, Sol spaces, Cartan-Vranceanu 3-dimensional spaces, or homogeneous contact 3-manifolds; we refer to [25][26][27].…”

confidence: 99%