Chen's conjecture on biharmonic submanifolds in Riemannian manifolds
arXiv:2108.10667
Abstract
We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian manifold. One of interesting problems in this direction is Chen's conjecture which says that any biharmonic submanifold in a Euclidean space is minimal. From the invariant equation for biharmonic submanifolds, we derive a fundamental identity involving the mean curvature vector field, and using this, we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space. More generally, it is proved that any biharmonic submanifold in a space form of nonpositively sectional curvature is minimal. Furthermore we provide affirmative partial answers to the generalized Chen's conjecture and Balmuş-Montaldo-Oniciuc conjecture.
We withdraw this paper because we found there are some missing steps in proofs