Biminimal properly immersed submanifolds in complete Riemannian manifolds of non-positive curvature

Abstract

We consider a non-negative biminimal properly immersed submanifold $M$ (that is, a biminimal properly immersed submanifold with $\lambda \ge 0$) in a complete Riemannian manifold $N$ with non-positive sectional curvature. Assume that the sectional curvature $K^N$ of $N$ satisfies $K^N\ge -L(1+{\mathrm{dist}}_N(\cdot ,q_0{)}^2{)}^{\frac{\alpha }{2}}$ for some $L>0,$ $2>\alpha \ge 0$ and $q_0\in N$. Then, we prove that $M$ is minimal. As a corollary, we give that any biharmonic properly immersed submanifold in a hyperbolic space is minimal. These results give affirmative partial answers to the global version of generalized Chen's conjecture.

11 pages

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