BIHARMONIC SUBMANIFOLDS OF ${\mathbb S}^3$

Caddeo, Renzo Ilario; Montaldo, Stefano; Oniciuc, Cezar

doi:10.1142/s0129167x01001027

Cited by 210 publications

(261 citation statements)

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“…are the main examples of proper-biharmonic submanifolds in S n (see [3,14]). Moreover, the following conjecture has been proposed.…”

Section: Biharmonic Maps ϕ : (M G) → (N H) Between Riemannian Manifmentioning

confidence: 99%

“…All proper-biharmonic submanifolds of S 2 and S 3 were determined (see [3,5]). The next step towards the classification of proper-biharmonic submanifolds in spheres is represented by the case of S 4 , and the first achievement was the proof of Conjecture 1.1 for compact hypersurfaces in S 4 (see [2]).…”

Section: Biharmonic Maps ϕ : (M G) → (N H) Between Riemannian Manifmentioning

confidence: 99%

See 1 more Smart Citation

Biharmonic Surfaces of S4

Balmus

Oniciuc

2009

Kyushu J. Math.

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Abstract. In this paper we prove that a constant mean curvature surface is proper-biharmonic in the unit Euclidean sphere S 4 if and only if it is minimal in a hypersphere S 3 (1/ √ 2). IntroductionBiharmonic maps ϕ : (M, g) → (N, h) between Riemannian manifolds are critical points of the bienergy functionalwhere τ (ϕ) = trace ∇ dϕ is the tension field of ϕ that vanishes for harmonic maps (see [10]). The Euler-Lagrange equation corresponding to E 2 is given by the vanishing of the bitension fieldwhere J ϕ is formally the Jacobi operator of ϕ (see [14]). The operator J ϕ is linear, thus any harmonic map is biharmonic. We call proper-biharmonic the non-harmonic biharmonic maps. The study of proper-biharmonic submanifolds, i.e. submanifolds such that the inclusion map is non-harmonic (non-minimal) biharmonic, constitutes an important research direction in the theory of biharmonic maps. The first ambient spaces considered for their properbiharmonic submanifolds were the spaces of constant sectional curvature. Non-existence results were obtained for proper-biharmonic submanifolds in Euclidean and hyperbolic spaces (see [1, 4, 7, 9, 12]).The case of the Euclidean sphere is different. Indeed, the hypersphere S n−1 (1/ √ 2) and the generalized Clifford torus (1/ √ 2), n 1 + n 2 = n − 1, n 1 = n 2 .2000 Mathematics Subject Classification: Primary 58E20.

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“…are the main examples of proper-biharmonic submanifolds in S n (see [3,14]). Moreover, the following conjecture has been proposed.…”

Section: Biharmonic Maps ϕ : (M G) → (N H) Between Riemannian Manifmentioning

confidence: 99%

Section: Biharmonic Maps ϕ : (M G) → (N H) Between Riemannian Manifmentioning

confidence: 99%

Biharmonic Surfaces of S4

Balmus

Oniciuc

2009

Kyushu J. Math.

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“…Thus,f satisfies a polynomial equation with constant coefficients, sof has to be a constant and then, f is a constant, i.e., grad f = 0 on U (in fact, f has to be zero). Therefore, we have a contradiction (see [6,8] for c = 0 and [3,4], for c = ±1).…”

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confidence: 96%

Global Properties of Biconservative Surfaces in $\mathbb{R}^3$ and $\mathbb{S}^3$

Oniciuc¹,

Nistor²

2017

Proceedings Book of International Workshop on Theory of Submanifolds

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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%