Biharmonic hypersurfaces in 4‐dimensional space forms

Balmus, Adina; Montaldo, Stefano; Oniciuc, Cezar

doi:10.1002/mana.200710176

Cited by 87 publications

(72 citation statements)

References 14 publications

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“…Biharmonic submanifolds have been studied by many geometers. For example, see [2], [3], [7], [8], [11], [12], [13], [14], [15], [18], [20], [21], [22], and the references therein. In a different setting, in [9], Chen defined a biharmonic submanifold M ⊂ E n of the Euclidean space as its mean curvature vector field H satisfies ∆H = 0 , where ∆ is the Laplacian.…”

Section: Let (M G) and (N H) Be 2 Riemannian Manifolds And F : (M mentioning

confidence: 99%

On Biharmonic Legendre curves in $S$-space forms

Özgür

Güvenç²

2014

Turk J Math

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Section: Let (M G) and (N H) Be 2 Riemannian Manifolds And F : (M mentioning

confidence: 99%

On Biharmonic Legendre curves in $S$-space forms

Özgür

Güvenç²

2014

Turk J Math

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“…Then, using certain results on isoparametric hypersurfaces, it was proved that there exist no compact properbiharmonic hypersurfaces of constant mean curvature and with three distinct principal curvatures everywhere in the unit Euclidean sphere (see [9]). …”

Section: Theorem ([8]) a Hypersurface With At Most Two Distinct Prinmentioning

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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“…(b) Any biharmonic hypersurfaces in H 4 (−1) is minimal (cf. [4]). (c) Any weakly convex biharmonic hypersurfaces in space form N m+1 (c) with c ≤ 0 is minimal (cf.…”

Section: Introductionmentioning

confidence: 99%

Some Results of Exponentially Biharmonic Maps Into a Non-Positively Curved Manifold

Han¹

2016

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we investigate exponentially biharmonic maps u : (M, g) → (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that ifthen u is harmonic. When u is an isometric immersion, we get that|H| q dvg < ∞ for 2 ≤ p < ∞ and 0 < q ≤ p < ∞, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N (c) with c ≤ 0 is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).

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Biharmonic hypersurfaces in 4‐dimensional space forms

Abstract: We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms

Cited by 87 publications

References 14 publications

On Biharmonic Legendre curves in $S$-space forms

On Biharmonic Legendre curves in $S$-space forms

Harmonic and Biharmonic Maps at Iaşi

Some Results of Exponentially Biharmonic Maps Into a Non-Positively Curved Manifold

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