Classification results for biharmonic submanifolds in spheres

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudo-umbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres.Comment: Dedicated to Professor Vasile Oproiu on his 65th birthday, 14 page

Properties of biharmonic submanifolds in spheres

In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.Comment: 10 pages; contribution to the Proceedings of the 11-th International Conference on Geometry, Integrability and Quantization, Varna 2009, Bulgari

A short survey on biharmonic maps between Riemannian manifolds

In this short survey we report on the theory of biharmonic maps between Riemannian manifolds.Comment: 20 page