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Some results of p-biharmonic submanifolds in a Riemannian manifold of non-positive curvature

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Abstract

In this paper, we introduce the notion of p-biharmonic submanifold. By using integral by parts, we obtain that any complete p-biharmonic submanifold (M, g) in a Riemannian manifold (N, h) with non-positive sectional curvature which satisfies an integral condition: for some \({q \in (0, \infty), \int_M |H|^q dv_g < \infty}\) must be minimal. This result gives affirmative partial answer to the conjecture 3 (generalized Chen’s conjecture for p-biharmonic submanifold). We also obtain that any p-biharmonic submanifold in a Riemannian manifold whose sectional curvature is smaller than \({-\varepsilon}\) for \({\varepsilon > 0}\) which satisfies that \({\int_{B_r (x_0)} |H|^{a + 2p - 2} dv_g (p \geq 2, a \geq 0)}\) is of at most polynomial growth of r, must be minimal.

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Correspondence to Yingbo Han.

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This work was supported by the National Natural Science Foundation of China (Grant No. 11201400) and Project for youth teacher of Xinyang Normal University (Grant No. 2014-QN-061.

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Han, Y. Some results of p-biharmonic submanifolds in a Riemannian manifold of non-positive curvature. J. Geom. 106, 471–482 (2015). https://doi.org/10.1007/s00022-015-0259-1

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