Chen’s conjecture on biharmonic submanifolds

Prove that every biharmonic submanifold in a Euclidean space is minimal, thereby resolving Chen’s conjecture.

Background

Biharmonic submanifolds generalize minimal submanifolds by requiring the mean curvature vector to be harmonic. In 1988, Chen proposed that biharmonicity in Euclidean space should force minimality. The paper presents this as a conjecture for which only partial affirmative results are known, including the result established here for four-dimensional pseudo-umbilical submanifolds in six-dimensional Euclidean space.

The conjecture remains broader than the paper’s main theorem because the paper treats pseudo-umbilical submanifolds of codimension two, rather than all biharmonic submanifolds in arbitrary dimensions and codimensions.

References

In 1988, Chen posed the following conjecture ``Any biharmonic submanifold in $\mathbb{E}{N}$ is minimal". This is known as Chen's conjecture .

— Four-dimensional pseudo-umbilical biharmonic submanifolds in the six-dimensional Euclidean space  (2609.17136 - Maeta, 15 Sep 2026) in Section 1, Introduction

However, the four-dimensional case has remained open for nearly four decades.

— Four-dimensional pseudo-umbilical biharmonic submanifolds in the six-dimensional Euclidean space  (2609.17136 - Maeta, 15 Sep 2026) in Section 1, Introduction

A long-standing problem in this direction is Chen's conjecture , which asserts that every biharmonic submanifold of a Euclidean space is minimal. Although the conjecture remains open in full generality, it has been confirmed for several important classes of submanifolds, including hypersurfaces in Euclidean spaces and submanifolds satisfying suitable curvature conditions; see, for example, .

— Biharmonic Lagrangian surfaces with constant Gaussian curvature in $2$-dimensional complex space forms  (2610.05703 - Fu et al., 5 Oct 2026) in Section 1, Introduction