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

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.