Balmuş–Montaldo–Oniciuc conjecture

Prove the Balmuş–Montaldo–Oniciuc conjecture that every proper biharmonic submanifold of a Euclidean sphere has constant mean curvature, together with its natural formulation for general space forms.

Background

The Balmuş–Montaldo–Oniciuc conjecture is motivated by the observation that most known examples of proper biharmonic submanifolds in Euclidean spheres have constant mean curvature. It proposes that constant mean curvature is forced for every proper biharmonic submanifold of a sphere.

The paper presents this as a major conjecture in biharmonic submanifold theory and notes that it can be formulated more generally in arbitrary space forms. The classification proved in the paper concerns a more specific class—biharmonic Lagrangian surfaces with constant Gaussian curvature in complex space forms—and does not resolve the conjecture in general.