Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature

Abstract

In this paper, we show that every biharmonic Wintgen ideal submanifold of dimension at least three or dimension two with codimension at most two in a Riemannian manifold of nonpositive constant sectional curvature is minimal. Under the same dimensional assumption, we also prove that every biharmonic Wintgen ideal submanifold in a Riemannian manifold of positive constant sectional curvature has constant mean curvature on each connected component. This gives partial affirmative answers to Chen's conjecture, to the generalized Chen's conjecture in hyperbolic spaces, and to the Balmuş-Montaldo-Oniciuc conjecture in spheres within the class of Wintgen ideal submanifolds.