In this note, we classify biharmonic submanifolds in a sphere defined by bieigenmaps (∆φ = λφ) or buckling eigenmaps (∆φ = −μ∆φ). The results can be viewed as generalizations of Takahashi’s characterization of minimal submanifolds in a sphere by eigenmaps. 1. Biharmonic maps and biharmonic subamnifolds Biharmonic maps are maps between Riemannian manifolds φ : (M, g) → (N, h) which are critical points of the bienergy functional

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)… Expand

We give some methods to construct examples of nonharmonic biharmonic submanifolds of the unitn-dimensional sphereSn. In the case of curves inSn we solve explicitly the biharmonic equation.

In this paper we consider eigenvalues of the Dirichlet biharmonic operator on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this… Expand

We obtain sharp lower bounds for the first eigenvalue of four types of eigenvalue problem defined by the bi-Laplace operator on compact manifolds with boundary and determine all the eigenvalues and… Expand

In this paper, we consider the Balmus-Montaldo-Oniciuc's conjecture in the case of hemispheres. We prove that a compact non-minimal biharmonic hypersurface in a hemisphere of $S^{n+1}$ must be the… Expand