Biharmonic hypersurfaces in complete Riemannian manifolds

Alı́as, Luis J.; García-Martínez, Sandra C.; Rigoli, Marco

doi:10.2140/pjm.2013.263.1

Cited by 17 publications

(9 citation statements)

References 19 publications

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“…• Complete biharmonic submanifolds with finite bi-energy and energy in a non-positively curved Riemannian manifold [149]. • Complete oriented biharmonic hypersurfaces M whose mean curvature H satisfying H ∈ L 2 (M ) in a Riemannian manifold with non-positive Ricci tensor [2]. • Compact biharmonic submanifolds in a Riemannian manifold with nonpositive sectional curvature [140].…”

Section: Recent Developments On Generalized Biharmonic Conjecturementioning

confidence: 99%

Some open problems and conjectures on submanifolds of finite type: recent development

2014

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Abstract. Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on this subject was given by the author in [48]. Recently, the study of finite type submanifolds, in particular, of biharmonic submanifolds, have received a growing attention with many progresses since the beginning of this century. In this article, we provide a detailed account of recent development on the problems and conjectures listed in [40].

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Section: Recent Developments On Generalized Biharmonic Conjecturementioning

confidence: 99%

Some open problems and conjectures on submanifolds of finite type: recent development

2014

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“…Since − → H is parallel, H is a constant. Using (7) and c > 0, we can simplify the second equation of (2) as…”

Section: Biharmonic Space-like Submanifolds With Parallel Mean Curvatmentioning

confidence: 99%

Minimality on biharmonic space-like submanifolds in pseudo-Riemannian space forms

Liu

Zhang

2015

Journal of Geometry and Physics

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“…During the past two decades, there has been a lot of research work done on biharmonic submanifolds in spheres or even in generic Riemannian manifolds (see, e.g., [7,[23][24][25][26][27][28][29][30][31]).…”

Section: Introductionmentioning

confidence: 99%

Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

Chen

2019

Mathematics

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The well known Chen’s conjecture on biharmonic submanifolds in Euclidean spaces states that every biharmonic submanifold in a Euclidean space is a minimal one. For hypersurfaces, we know from Chen and Jiang that the conjecture is true for biharmonic surfaces in E 3 . Also, Hasanis and Vlachos proved that biharmonic hypersurfaces in E 4 ; and Dimitric proved that biharmonic hypersurfaces in E m with at most two distinct principal curvatures. Chen and Munteanu showed that the conjecture is true for δ ( 2 ) -ideal and δ ( 3 ) -ideal hypersurfaces in E m . Further, Fu proved that the conjecture is true for hypersurfaces with three distinct principal curvatures in E m with arbitrary m. In this article, we provide another solution to the conjecture, namely, we prove that biharmonic surfaces do not exist in any Euclidean space with parallel normalized mean curvature vectors.

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Biharmonic hypersurfaces in complete Riemannian manifolds

Abstract: We consider biharmonic hypersurfaces in complete Riemannian manifolds and prove that, under some additional assumptions, they are minimal.

Cited by 17 publications

References 19 publications

Some open problems and conjectures on submanifolds of finite type: recent development

Some open problems and conjectures on submanifolds of finite type: recent development

Minimality on biharmonic space-like submanifolds in pseudo-Riemannian space forms

Chen’s Biharmonic Conjecture and Submanifolds with Parallel Normalized Mean Curvature Vector

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