Biharmonic submanifolds with parallel mean curvature vector field in spheres

TL;DR

This paper investigates the properties and classifications of proper biharmonic submanifolds in spheres, focusing on those with parallel mean curvature vectors and Weingarten operators, providing new classification results.

Contribution

It offers a partial classification and a complete classification of proper biharmonic submanifolds with parallel mean curvature vectors in spheres, including conditions involving the Weingarten operator.

Findings

Boundedness results for mean curvature in proper biharmonic submanifolds

Partial classification of such submanifolds with parallel mean curvature vectors

Complete classification when the Weingarten operator is parallel

Abstract

We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.