Biharmonic submanifolds with parallel mean curvature vector field in spheres

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.2010 Mathematics Subject Classification. 58E20.

Biharmonic submanifolds into ellipsoids

“…The Chen's conjecture was generalized in [7] for biharmonic submanifolds into a Riemannian manifold with non-positive sectional curvature, although Y. Ou and L. Tang found in [14] a counterexample. These facts have pushed research towards the investigation of biharmonic submanifolds of the Euclidean sphere (see [1,2,3,4,5,7,6] for an overview of the main results in this context). A further step is the study of biharmonic submanifolds into Euclidean ellipsoids, because these manifolds are geometrically rich and interestingly do not have constant sectional curvature: in [13] we obtained a complete classification of proper biharmonic curves into 3-dimensional ellipsoids and, more generally, into any non-degenerate quadric.…”

Some recent progress of biharmonic submanifolds

“…• Any proper biharmonic surface in S 3 is a part of S 2 ( 1 √ 2 ) (Caddeo-Montaldo-Oniciuc [14]). [4] and [12]…”

Biharmonic submanifolds in manifolds with bounded curvature

“…Balmus and Oniciuc also showed that (cf. [9]) : Let M be a compact non-minimal biharmonic submanifold of S n . Then either (i) there exists a point p ∈ M such that |H(p)| < 1, (ii) |H| = 1.…”