Biharmonic hypersurfaces in Riemannian manifolds

By Ye-lin Ou. 2009

139 citations recorded by the source corpus.

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied by Jiang, Chen, Caddeo, Montaldo, and Oniciuc. We then apply the equation to show that the generalized Chen conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a 2-parameter family of conformally flat metrics and a 4-parameter family of multiply warped product metrics, each of which turns the foliation of an upper-half space of R m by parallel hyperplanes into a foliation with each leaf a proper biharmonic hypersurface. We also study the biharmonicity of Hopf cylinders of a Riemannian submersion.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b7d434afa13307dcf786c0bceaa9a8ea2eaba81a

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

Some open problems and conjectures on submanifolds of finite type: recent development (2014, background): “• Totally umbilical biharmonic hypersurfaces in Einstein spaces [151].”
On the generalized Chen's conjecture on biharmonic submanifolds (2010, methodology): “, or (3) ai = 0, i= 1, ··· , mand f(z) = 1 Az+B , where Aand Bare constants. In this case each hyperplane is a proper biharmonic hypersurface. This recovers a result (Theorem 3.1) obtained earlier in [Ou1]. Proof. We have σ= lnf(z).Using the notations ∂i = ∂ ∂xi ,i= 1,2,...,m,∂m+1 = ∂ ∂z , we can ea”
Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature (2011, unknown): “[1], [2], [3], [4], [10], [11], [12]) is that: 2000 Mathematics Subject Classification.”
Proper biconservative immersions into the Euclidean space (2013, methodology): “The decomposition of the bitension field with respect to its normal and tangent components was obtained with contributions of [3,7,23,25,26], and for hypersurfaces, it can be summarized in the following theorem.”
Biharmonic submanifolds of $${\mathbb{C}P^n}$$ (2009, background): “ensively studied, see, for example, [3, 4, 5, 6, 8, 11]. Naturally, the next step has been the study of proper-biharmonic submanifolds of spaces of non-constant sectional curvature (see, for example, [1, 13, 14, 15, 25, 26, 27]). This work is devoted to the study of proper-biharmonic submanifolds in”
Biharmonic ideal hypersurfaces in Euclidean spaces (2013, background): “In particular, there exist many results on the generalized Chen’s conjecture (see, for instance, [1–4,26–30]).”
Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces (2009, unknown): “etric immersions: (i) The generalized Cliﬀord torus Sp(√1 2 )×Sq(√1 2 ) → Snwith p6= q,p+q= n([18]); (ii) The hypersphere Sn(√1 2 ) → Sn+1 ([6]); (iii) Biharmonic cylinder : S1(√1 2 ) × R→ S2 × R ([27]); (iv) The hyperplanes z = k (k is a constant > −C) in the conformally ﬂat space (R m+1 + ,h= (”
Biharmonic PNMC submanifolds in spheres (2011, background): “This result was obtained in [10, 25] for submanifolds in space forms, and in [26] for general hypersurfaces.”