Biharmonic submanifolds in manifolds with bounded curvature

Abstract

Abstract. We consider a complete biharmonic submanifold φ : (M, g) → (N, h) in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant c. Assume that the mean curvature is bounded from below byfor some 0 < p < ∞, or (ii) the Ricci curvature of M is bounded from below, then the mean curvature is √ c. Furthermore, if M is compact, then we obtain the same result without the assumption (i) or (ii). IntroductionIn 1983, J. Eells and L. Lemaire [16] proposed the problem to consider biharmonic maps. Biharmonic maps are, by definition, a generalization of harmonic maps. As well known, harmonic maps have been applied into various fields in differential geometry. In 1964, J. Eells and J. H. Sampson considered the existence problem of harmonic maps between compact Riemannian manifolds. They showed that any continuous map from a compact Riemannian manifold into a compact Riemannian manifold of non-positive curvature is free homotopically deformable to harmonic maps. By using the existence and properties of harmonic maps, one can study the structure of Riemannian manifolds. On the other hand, non-existence results for harmonic maps are also known. For example, a map of degree ±1 from a 2-dimensional torus into a 2-dimensional sphere is not homotopic to any harmonic map. Therefore a generalization of harmonic maps seems an important subject instead. So far, it seems a biharmonic map. We would 2010 Mathematics Subject Classification. primary 58E20, secondary 53C43.

Biharmonic hypersurfaces with bounded mean curvature

“…The author reformulated BMO conjecture into a problem in global differential geometry (cf. [13] (ii) An orientable Dupin hypersurface (cf. [1]).…”