We study biharmonic maps between Riemannian manifolds with finite energy and finite bi-energy. We show that if the domain is complete and the target of non-positive curvature, then such a map is harmonic. We then give applications to isometric immersions and horizontally conformal submersions.2000 Mathematics Subject Classification. primary 58E20, secondary 53C43.
In this article, we show that, for a biharmonic hypersurfacewhere H is the mean curvature of (M, g) in (N , h), then (M, g) is minimal in (N , h). Thus, for a counter example (M, g) in the case of hypersurfaces to the generalized Chen's conjecture (cf. Sect. 1), it holds that M |H | 2 v g = ∞.
In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if M |η| 2 v g < ∞, then (M, g) is minimal in (N, h), i.e., η ≡ 0, where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition M |η| 2 v g < ∞ to the following generalized B.Y. Chen's conjecture: every biharmonic submanifold of a Riemannian manifold with non-positive sectional curvature must be minimal. The conjecture turned out false in case of an incomplete Riemannian manifold (M, g) by a counter example of Y-L. Ou and L. Tang [12] 2000 Mathematics Subject Classification. 58E20.
A triharmonic map is a critical point of the 3-energy in the space of smooth maps between two Riemannian manifold. We study the generalized Chen's conjecture for a triharmonic isometric immersion ϕ into a space form of non-positively constant curvature. We show that if the domain is complete and both the 4-energy of ϕ, and the L 4 -norm of the tension field τ (ϕ), are finite, then such an immersion ϕ is minimal.2000 Mathematics Subject Classification. primary 58E20, secondary 53C43.
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