The index of biharmonic maps in spheres

Loubeau, E.; Oniciuc, Cezar

doi:10.1112/s0010437x04001204

Cited by 36 publications

(36 citation statements)

References 12 publications

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“…The second variation formula for biharmonic maps in spheres was deduced [26] and the stability of certain classes of biharmonic maps in spheres was discussed in [23,24]. Also, in [31] there were given some sufficient conditions for the instability of Legendre proper biharmonic submanifolds in Sasakian space forms and the author proved the instability of Legendre curves and surfaces in Sasakian space forms.…”

Section: Introductionmentioning

confidence: 99%

Biharmonic maps between warped product manifolds

Balmus

Montaldo

Oniciuc

2007

Journal of Geometry and Physics

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Biharmonic maps between warped products are studied. The main results are:\ud (i) the condition for the biharmonicity of the inclusion of a Riemannian manifold N into the warped product M ×f 2 theprojectionπ : M×f2 N → M;\ud N and of\ud (ii) the construction of two new classes of non-harmonic biharmonic maps using products of harmonic maps φ = 1M × ψ : M × N → M × N and warping the metric on their domain or codomain;\ud (iii) the study of three classes of axially symmetric biharmonic maps, using the warped product setting

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Section: Introductionmentioning

confidence: 99%

Biharmonic maps between warped product manifolds

Balmus

Montaldo

Oniciuc

2007

Journal of Geometry and Physics

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“…) → S n+1 was computed in [13], and it is exactly one. Then they were investigated the indices of biharmonic maps in the unit Euclidean sphere S n+1 obtained from minimal Riemannian immersions in S n (…”

Section: Introductionmentioning

confidence: 99%

On the biharmonic and harmonic indices of the Hopf map

Loubeau

Oniciuc

2007

Trans. Amer. Math. Soc.

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Abstract. Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalize harmonic maps. We consider the Hopf map ψ : S 3 → S 2 and modify it into a nonharmonic biharmonic map φ : S 3 → S 3 . We show φ to be unstable and estimate its biharmonic index and nullity. Resolving the spectrum of the vertical Laplacian associated to the Hopf map, we recover Urakawa's determination of its harmonic index and nullity.

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“…The second variation for the bienergy functional was first studied in a general setting in [38] and then concrete results on the stability of biharmonic maps to spheres were obtained (see [42,43,51]). Since biharmonic Riemannian immersions in spheres are stable, i.e.…”

Section: Conjecture ([8]) Any Proper-biharmonic Submanifold In S N Hmentioning

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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The index of biharmonic maps in spheres

Cited by 36 publications

References 12 publications

Biharmonic maps between warped product manifolds

Biharmonic maps between warped product manifolds

On the biharmonic and harmonic indices of the Hopf map

Harmonic and Biharmonic Maps at Iaşi

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