Abstract. We consider a complete biharmonic hypersurface with nowhere zero mean curvature vector field φ : (M m , g) → (S m+1 , h) in a sphere. If the squared norm of the second fundamental form B is bounded from above by m, and M H −p dv g < ∞, for some 0 < p < ∞, then the mean curvature is constant.
IntroductionThe problem of biharmonic maps was suggested in 1964 by J. Eells and J. H. Sampson (cf. [6]). Biharmonic maps are generalizations of harmonic maps. As well known, harmonic maps have been applied into various fields in differential geometry. However there are non-existence results for harmonic maps. Therefore a generalization of harmonic maps is an important subject.G. Y. Jiang [8] considered a biharmonic submanifold, and gave some examples of non-minimal biharmonic submanifolds in S n as follows: (i)There are many studies of biharmonic submanifolds in spheres. Interestingly, their studies suggest the following BMO conjecture which was introduced by Balmus, Montaldo and Oniciuc (cf. [2]).Conjecture 1 (BMO conjecture). Any biharmonic submanifold in spheres has constant mean curvature.2010 Mathematics Subject Classification. primary 53C43, secondary 58E20, 53C40.