On the biharmonic and harmonic indices of the Hopf map

Loubeau, E.; Oniciuc, Cezar

doi:10.1090/s0002-9947-07-03934-7

Cited by 57 publications

(37 citation statements)

References 21 publications

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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%

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

confidence: 99%

“…Theorem ( [43]). The index of the proper-biharmonic subimmersion ϕ : S 3 ( √ 2) → S 3 , which is derived from the usual harmonic Hopf map ψ : S 3 ( √ 2) → S 2 (1/ √ 2), is at least 11.…”

Section: Corollary ([42]) the Biharmonic Map Derived From The Generamentioning

confidence: 99%

Harmonic and Biharmonic Maps at Iaşi

Balmus¹,

Fetcu²,

Oniciuc³

2010

Annals of the Alexandru Ioan Cuza University - Mathematics

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“…[2]∼ [12], [15], [17], [18], [19], [22], [24], [32], etc...). Interestingly, their examples and classification results suggest that "any biharmonic submanifold in spheres has constant mean curvature".…”

Section: Introductionmentioning

confidence: 99%

“…Letting r ր ∞ in (24), the right hand side of (24) goes to zero and the left hand side of (24) goes to…”

mentioning

confidence: 99%

Biharmonic submanifolds in manifolds with bounded curvature

Maeta

2016

Int. J. Math.

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Abstract. We consider a complete biharmonic submanifold φ : (M, g) → (N, h) in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant c. Assume that the mean curvature is bounded from below byfor some 0 < p < ∞, or (ii) the Ricci curvature of M is bounded from below, then the mean curvature is √ c. Furthermore, if M is compact, then we obtain the same result without the assumption (i) or (ii). IntroductionIn 1983, J. Eells and L. Lemaire [16] proposed the problem to consider biharmonic maps. Biharmonic maps are, by definition, a generalization of harmonic maps. As well known, harmonic maps have been applied into various fields in differential geometry. In 1964, J. Eells and J. H. Sampson considered the existence problem of harmonic maps between compact Riemannian manifolds. They showed that any continuous map from a compact Riemannian manifold into a compact Riemannian manifold of non-positive curvature is free homotopically deformable to harmonic maps. By using the existence and properties of harmonic maps, one can study the structure of Riemannian manifolds. On the other hand, non-existence results for harmonic maps are also known. For example, a map of degree ±1 from a 2-dimensional torus into a 2-dimensional sphere is not homotopic to any harmonic map. Therefore a generalization of harmonic maps seems an important subject instead. So far, it seems a biharmonic map. We would 2010 Mathematics Subject Classification. primary 58E20, secondary 53C43.

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“…These vector fields verify that ∇ μ V μ C 2s = (μ − 2s)/ |μ| JC 2s and they allow us to prove in Section 3 that on Lorentzian Berger spheres, the Hopf vector fields V μ are unstable critical points of the energy, the volume and the generalized energy E g λ for all λ < 0. The eigenfunctions of Δ v have been also used to study, for example, the harmonic index and nullity of the Hopf map (see [14]). …”

Section: Introductionmentioning

confidence: 99%

Instability of Hopf vector fields on Lorentzian Berger spheres

Hurtado

2010

Isr. J. Math.

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In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.

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On the biharmonic and harmonic indices of the Hopf map

Cited by 57 publications

References 21 publications

Harmonic and Biharmonic Maps at Iaşi

Harmonic and Biharmonic Maps at Iaşi

Biharmonic submanifolds in manifolds with bounded curvature

Instability of Hopf vector fields on Lorentzian Berger spheres

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