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arXiv · 0901.1507

Biharmonic hypersurfaces in Riemannian manifolds

Ye-Lin Ou
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Abstract

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 in \cite{Ji2}, \cite{CH}, \cite{CMO1}, \cite{CMO2}. We then apply the equation to show that the generalized Chen's 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 $\mathhbb{R}^m$ by parallel hyperplanes into a foliation with each leave a proper biharmonic hypersurface. We also characterize proper biharmonic vertical cylinders in $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$.

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Ye-Lin Ou. 2009-12-09. Biharmonic hypersurfaces in Riemannian manifolds. https://arxiv.org/abs/0901.1507

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