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 ℝm
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.
Keywords
biharmonic maps, biharmonic hypersurfaces, biharmonic
foliations, conformally flat space, Einstein space
