Biharmonic Riemannian submersions from 3-manifolds

An important theorem about biharmonic submanifolds proved independently by Chen-Ishikawa [CI] and Jiang [Ji] states that an isometric immersion of a surface into 3-dimensional Euclidean space is biharmonic if and only if it is harmonic (i.e, minimal). In a later paper [CMO2], Cadeo-Monttaldo-Oniciuc shown that the theorem remains true if the target Euclidean space is replaced by a 3-dimensional hyperbolic space form. In this paper, we prove the dual results for Riemannian submersions, i.e., a Riemannian submersion from a 3-dimensional space form of non-positive curvature into a surface is biharmonic if and only if it is harmonic

Biharmonic PNMC Submanifolds in Spheres

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in $\mathbb{S}^n$. Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in $\mathbb{S}^n$, their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic submanifold in $\mathbb{S}^n$ is of 1-type or 2-type if and only if it has constant mean curvature {\mcf}=1 or {\mcf}\in(0,1), respectively; (ii) there are no proper biharmonic 3-type submanifolds with parallel normalized mean curvature vector field in $\mathbb{S}^n$.Comment: 17 page

Differential inequalities on complete Riemannian manifolds and applications

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46226/1/208_2005_Article_BF01455859.pd