Abstract
In this paper, we show that, for every biharmonic submanifold (M, g) of a Riemannian manifold (N, h) with non-positive sectional curvature, if \({\int_M\vert \eta \vert^2 v_g < \infty}\) , then (M, g) is minimal in (N, h), i.e., \({\eta\equiv0}\), where η is the mean curvature tensor field of (M, g) in (N, h). This result gives an affirmative answer under the condition \({\int_M\vert \eta \vert^2 v_g < \infty}\) to the following generalized 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 Ou and Tang (in The generalized Chen’s conjecture on biharmonic sub-manifolds is false, a preprint, 2010).
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H. Urakawa’s research is supported by the Grant-in-Aid for the Scientific Research, (C), No. 21540207, Japan Society for the Promotion of Science.
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Nakauchi, N., Urakawa, H. Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature. Results. Math. 63, 467–474 (2013). https://doi.org/10.1007/s00025-011-0209-7
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DOI: https://doi.org/10.1007/s00025-011-0209-7