Biharmonic PNMC Submanifolds in Spheres

Abstract

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 ${\mathbb{S}}^n$.

17 pages

References in corpus (2)

• Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries
• Biharmonic surfaces of ${\mathbb{S}}^4$

Cited by in corpus (22)

• On f-biharmonic maps and f-biharmonic submanifolds
• Biharmonic maps from a 2-sphere
• On cohomogeneity one biharmonic hypersurfaces into the Euclidean space
• Complete biconservative surfaces in ${\mathbb{R}}^3$ and ${\mathbb{S}}^3$
• On the Existence of Closed Biconservative Surfaces in Space Forms
• Closed Biconservative Hypersurfaces in Spheres
• Biharmonic surfaces with parallel mean curvature in complex space forms
• Reduction methods for the bienergy
• Biharmonic hypersurfaces with constant scalar curvature in space forms
• On Chen's biharmonic conjecture for hypersurfaces in ${\mathbb{R}}^5$
• Complete classification of biconservative hypersurfaces with diagonalizable shape operator in Minkowski 4-space
• On biharmonic submanifolds in non-positively curved manifolds
• Biharmonic submanifolds into ellipsoids
• Biharmonic submanifolds of pseudo-Riemannian manifolds
• Biharmonic maps from tori into a 2-sphere
• Complete biconservative surfaces in the hyperbolic space ${\mathbb{H}}^3$
• Proper Biconservative immersions into the Euclidean space
• CMC biconservative surfaces in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$
• Chen's conjecture on biharmonic submanifolds in Riemannian manifolds
• Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor
• Biharmonic hypersurfaces in a product space $L^m\times \mathbb{R}$
• CMC proper-biharmonic surfaces of constant Gaussian curvature in spheres