Classification results for biharmonic submanifolds in spheres
TLDR
In this article, Chen et al. studied the rigidity of pseudoumbilical biharmonic submanifolds of codimension 2 and for B-Y surfaces with parallel mean curvature vector field.Abstract:
We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.read more
Citations
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Biharmonic hypersurfaces in Riemannian manifolds
TL;DR: In this article, the generalized Chen conjecture is proven to be true for totally umbilical biharmonic hypersurfaces in an Einstein space, and a 2-parameter family of conformally flat metrics and a 4-parameters family of multiply warped product metrics, each of which turns the foliation of an upper-half space of a Riemannian manifold by parallel hyperplanes into a foliation with each leaf a proper hypersurface.
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Biharmonic submanifolds of \({\mathbb{C}P^n}\)
TL;DR: In this article, the relation between the bitension field of the inclusion of a submanifold and the Hopf-tube in a complex space form was studied and a new family of proper-biharmonic sub-manifolds of the complex projective space was given.
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Biharmonic hypersurfaces in 4-dimensional space forms
TL;DR: In this paper, the authors focus their attention on bi-harmonic submanifolds such that the inclusion map is a biharmonic map, i.e. a non-minimal bi-harmonic map with constant sectional curvature.
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Some open problems and conjectures on submanifolds of finite type: recent development
TL;DR: In this article, a detailed account of recent development on the problems and conjectures listed in [40] is given. And a detailed survey of the results up to 1996 was given by the author in [48].
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Biharmonic ideal hypersurfaces in Euclidean spaces
TL;DR: In this paper, it was shown that the bi-harmonic conjecture is true for δ ( 2 ) -ideal and δ( 3 )-ideal hypersurfaces of a Euclidean space of arbitrary dimension.
References
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MonographDOI
Total Mean Curvature and Submanifolds of Finite Type
TL;DR: Differentiable Manifolds Riemannian and Pseudo-Riemannians Manifold Theory and Spectral Geometry Submanifolds Total Mean Curvature Submannifolds of Finite Type Biharmonic and Bihharmonic Conjectures as discussed by the authors.
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Biharmonic submanifolds in spheres
TL;DR: In this paper, the authors give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphere, where the bi-harmonic equation is solved explicitly.