Biharmonic properly immersed submanifolds in Euclidean spaces

Akutagawa, Kazuo; Maeta, Shun

doi:10.1007/s10711-012-9778-1

Cited by 67 publications

(59 citation statements)

References 9 publications

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“…Only partial answers to Chen's conjecture have been obtained for more than three decades, e.g. [1], [2], [10], [32]. In the case of hypersurfaces, Chen's conjecture is true for the following special cases:…”

Section: Introductionmentioning

confidence: 99%

Biharmonic hypersurfaces with constant scalar curvature in space forms

Hong

2018

Pacific J. Math.

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Let M n be a biharmonic hypersurface with constant scalar curvature in a space form M n+1 (c). We show that M n has constant mean curvature if c > 0 and M n is minimal if c ≤ 0, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space E n+1 or hyperbolic space H n+1 for n < 7.

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Section: Introductionmentioning

confidence: 99%

Biharmonic hypersurfaces with constant scalar curvature in space forms

Hong

2018

Pacific J. Math.

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“…K. Akutagawa and the author showed that biharmonic properly immersed submanifold in the Euclidean space is minimal [1]. Here, we remark that the properness of the immersion implies the completeness of (M, g).…”

Section: Introductionmentioning

confidence: 65%

“…Recently, we reformulated Conjecture 1 into a problem in global differential geometry as the following (cf. [1,14,15]):…”

Section: Introductionmentioning

confidence: 99%

Biminimal properly immersed submanifolds in the Euclidean spaces

Maeta

2012

Journal of Geometry and Physics

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Abstract. We consider a complete nonnegative biminimal submanifold M (that is, a complete biminimal submanifold with λ ≥ 0) in a Euclidean space E N . Assume that the immersion is proper, that is, the preimage of every compact set in E N is also compact in M . Then, we prove that M is minimal. From this result, we give an affirmative partial answer to Chen's conjecture. For the case of λ < 0, we construct examples of biminimal submanifolds and curves.

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“…K. Akutagawa and S. Maeta [3] gave a remarkable breakthrough to Chen's conjecture, by giving the following answer to this conjecture in the case of properly-immersed submanifolds of the Euclidean space:…”

Section: Chen's Conjecture and The Generalized Chen's Conjecturementioning

confidence: 99%