Chen's conjecture on biharmonic submanifolds in Riemannian manifolds

TL;DR

This paper proves Chen's conjecture that biharmonic submanifolds in Euclidean space are minimal, and extends results to space forms with nonpositive curvature, providing partial answers to related conjectures.

Contribution

It establishes the validity of Chen's conjecture in Euclidean spaces and space forms with nonpositive curvature, and offers partial solutions to generalized conjectures.

Findings

Chen's conjecture is confirmed for Euclidean spaces.

Biharmonic submanifolds in nonpositively curved space forms are minimal.

Partial affirmative results are provided for related conjectures.

Abstract

We study biharmonic hypersurfaces and biharmonic submanifolds in a Riemannian manifold. One of interesting problems in this direction is Chen's conjecture which says that any biharmonic submanifold in a Euclidean space is minimal. From the invariant equation for biharmonic submanifolds, we derive a fundamental identity involving the mean curvature vector field, and using this, we prove Chen's conjecture on biharmonic submanifolds in a Euclidean space. More generally, it is proved that any biharmonic submanifold in a space form of nonpositively sectional curvature is minimal. Furthermore we provide affirmative partial answers to the generalized Chen's conjecture and Balmu\c{s}-Montaldo-Oniciuc conjecture.