Biharmonic submanifolds of $${\mathbb{C}P^n}$$

Fetcu, Dorel; Loubeau, E.; Montaldo, Stefano; Oniciuc, Cezar

doi:10.1007/s00209-009-0582-z

Cited by 44 publications

(27 citation statements)

References 19 publications

(31 reference statements)

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“…Results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space were also proved (see [28]). There was obtained the relation between the bitension field of the inclusion ȷ : M → CP n of a submanifold in CP n and the bitension field of the inclusion ȷ : M → S 2n+1 of the corresponding Hopf cylinder in S 2n+1 ,…”

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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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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“…There exist many non-minimal biharmonic submanifolds in a sphere or a complex projective space (see, for example, [5] and [25]). On the other hand, the following conjecture proposed by Chen [12] is still open.…”

Section: Ideal Cr Immersions As Critical Points Of λ-Bienergy Functionalmentioning

confidence: 99%

Ideal CR Submanifolds

Sasahara

2016

Geometry of Cauchy-Riemann Submanifolds

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This paper surveys some of the known results on δ-ideal CR submanifolds in complex space forms, the nearly Kähler 6-sphere and odd dimensional unit spheres. In addition, the relationship between δ-ideal CR submanifolds and critical points of the λ-bienergy is mentioned. Some topics on variational problem for the λ-bienergy are also presented.2010 Mathematics Subject Classification. 53C42, 53B25.Let M be an n-dimensional submanifold of a Riemannian manifoldM . Let us denote by ∇ and∇ the Levi-Civita connections on M andM , respectively. The Gauss and Weingarten formulas are respectively given bỹ

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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%

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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Biharmonic submanifolds of $${\mathbb{C}P^n}$$

Cited by 44 publications

References 19 publications

Harmonic and Biharmonic Maps at Iaşi

Harmonic and Biharmonic Maps at Iaşi

Ideal CR Submanifolds

Biharmonic submanifolds in manifolds with bounded curvature

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