2009
DOI: 10.2206/kyushujm.63.339
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Biharmonic Surfaces of S4

Abstract: 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 f… Show more

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Cited by 11 publications
(6 citation statements)
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References 15 publications
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“…Remark 4.6. If n = 4 in Theorem 4.5, then the same conclusion holds under the weakened assumption that the surface is CMC as it was shown in [7].…”
Section: Pmc Biharmonic Immersions In S Nsupporting
confidence: 60%
See 1 more Smart Citation
“…Remark 4.6. If n = 4 in Theorem 4.5, then the same conclusion holds under the weakened assumption that the surface is CMC as it was shown in [7].…”
Section: Pmc Biharmonic Immersions In S Nsupporting
confidence: 60%
“…The goal of this paper is to continue the study of proper biharmonic submanifolds in S n in order to achieve their classification. This program was initiated for the very first time in [26] and then developed in [1] - [7], [9,10,29,30,32].…”
Section: Introductionmentioning
confidence: 99%
“…The Chen's conjecture was generalized in [7] for biharmonic submanifolds into a Riemannian manifold with non-positive sectional curvature, although Y. Ou and L. Tang found in [14] a counterexample. These facts have pushed research towards the investigation of biharmonic submanifolds of the Euclidean sphere (see [1,2,3,4,5,7,6] for an overview of the main results in this context). A further step is the study of biharmonic submanifolds into Euclidean ellipsoids, because these manifolds are geometrically rich and interestingly do not have constant sectional curvature: in [13] we obtained a complete classification of proper biharmonic curves into 3-dimensional ellipsoids and, more generally, into any non-degenerate quadric.…”
Section: Introductionmentioning
confidence: 99%
“…Thus ϕ is a 2-type submanifold of R 6 with eigenvalues 2 and 4. It is easy to verify that ϕ is a [2,4]-order immersion in R 6 .…”
Section: Other Examples Of Proper Biharmonic Immersed Submanifolds Inmentioning
confidence: 98%