Biharmonic maps between warped product manifolds

Balmus, Adina; Montaldo, Stefano; Oniciuc, Cezar

doi:10.1016/j.geomphys.2006.03.012

Cited by 34 publications

(38 citation statements)

References 21 publications

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“…The last part of this paper is devoted to the study of biharmonic maps between warped product manifolds, these maps were also studied in [17]. We will give some results of the biharmonicity in other particular cases.…”

Section: Remark 4 Looking For Particular Solutions Of Typementioning

confidence: 99%

Conformal Maps, Biharmonic Maps, and the Warped Product

Ouakkas

Djebbouri

2016

Mathematics

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Section: Remark 4 Looking For Particular Solutions Of Typementioning

confidence: 99%

Conformal Maps, Biharmonic Maps, and the Warped Product

Ouakkas

Djebbouri

2016

Mathematics

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“…(A) One can check (see also [BMO2]) that for the cases (i) p = 0, and (ii) p = m, the maps φ = x |x| p are actually harmonic maps. We know that in these cases these maps are f -biharmonic for any f .…”

Section: The Bi-laplacian Of U In Other Words a Function U Is An F mentioning

confidence: 99%

“…In this case, (iii) and (iv) imply that φ = x |x| p is a proper biharmonic map if and only if p = −2, or p = m − 2. Note that the case p = −2 was missed in the list of Remark 5.8 in [BMO2]. (B) For p = 0, m, and k = 0, the maps in cases (iii) and (iv) provide infinitely many example of proper f -biharmonic maps (i.e., which are neither harmonic nor biharmonic maps).…”

Section: The Bi-laplacian Of U In Other Words a Function U Is An F mentioning

confidence: 99%

“…Since 2000, biharmonic maps have been receiving a growing attention and have become a popular subject of study with many progresses. For some recent geometric study of general biharmonic maps see [BK], [BFO2], [BMO2], [MO], [NUG], [Ou1], [Ou4], [OL], [Oua] and the references therein. For some recent study of biharmonic submanifolds see [CI], [Ji2], [Ji3], [Di], [CMO1], [CMO2], [BMO1], [BMO3], [Ou3], [OT], [OW], [NU], [TO], * [CM], [AGR] and the references therein.…”

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confidence: 99%

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Onf-biharmonic maps andf-biharmonic submanifolds

2014

Pacific J. Math.

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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 derive f -biharmonic submanifolds equations and prove that a surface in a manifold (N n , h) is an fbiharmonic 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.

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“…New examples of proper-biharmonic maps were also obtained by using warped product manifolds (see [7]). …”

Section: H) Being the Corresponding Structures If ϕ : M → N Is A Smomentioning

confidence: 99%

Harmonic and Biharmonic Maps at Iaşi

Balmus¹,

Fetcu²,

Oniciuc³

2010

Annals of the Alexandru Ioan Cuza University - Mathematics

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Abstract. We report on the achievements of the geometers from Iaşi in the field of harmonic and biharmonic maps.Mathematics Subject Classification 2000: 53C07, 53C40, 53C42, 53C43, 58E20. Key words: harmonic and biharmonic maps, minimal and biharmonic submanifolds.Nowadays, the theory of harmonic maps between Riemannian manifolds is a very important field of Riemannian geometry. The harmonic maps ϕ : (M, g) → (N, h) are critical points of the energy functional E which is defined on the infinit dimensional manifold of the smooth maps from M to N ,Here dϕ is the differential of the map ϕ and ∥·∥ denotes the Hilbert-Schmidt norm. The Euler-Lagrange equation associated to the energy E is given by the vanishing of the tension field τ (ϕ) = trace ∇dϕ and it is a second order elliptic equation, as one can see from its local expressionwhere Γ denotes the Christoffel symbols and ∆ is the Laplacian acting on functions. From here raises the deep link established by the harmonic maps *

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Biharmonic maps between warped product manifolds

Cited by 34 publications

References 21 publications

Conformal Maps, Biharmonic Maps, and the Warped Product

Conformal Maps, Biharmonic Maps, and the Warped Product

Onf-biharmonic maps andf-biharmonic submanifolds

Harmonic and Biharmonic Maps at Iaşi

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