Biharmonic PNMC Submanifolds in Spheres

TL;DR

This paper classifies biharmonic submanifolds in spheres with parallel normalized mean curvature, determines all such surfaces, and analyzes their type, revealing specific conditions under which they are of 1-type or 2-type.

Contribution

It provides a complete classification of biharmonic submanifolds with parallel normalized mean curvature in spheres and characterizes their type based on mean curvature values.

Findings

Biharmonic submanifolds with parallel normalized mean curvature are classified.

All biharmonic surfaces with this property in spheres are determined.

Proper biharmonic submanifolds are of 1-type or 2-type depending on mean curvature.

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 $\mathbb{S}^n$.