Properly immersed submanifolds in complete Riemannian manifolds

We consider a properly immersed submanifold M in a complete Riemannian manifold N  . Assume that the sectional curvature KNKN of N   satisfies KN⩾−L(1+distN(⋅,q0)2)α2 for some L>0L>0, 2>α⩾02>α⩾0 and q0∈Nq0∈N. If there exists a positive constant k>0k>0 such that Δ|H|2⩾k|H|4Δ|H|2⩾k|H|4, then we prove that M is minimal. We also obtain similar results for totally geodesic submanifolds. Furthermore, we consider a properly immersed submanifold M in a complete Riemannian manifold N   with KN⩾−L(1+distN(⋅,q0)2)α2 for some L>0L>0, 2>α⩾02>α⩾0 and q0∈Nq0∈N. Let u be a smooth non-negative function on M  . If there exists a positive constant k>0k>0 such that Δu⩾ku2Δu⩾ku2, and |H|⩽C(1+distN(⋅,q0)2)β2 for some C>0C>0 and 1>β⩾01>β⩾0, then we prove that u=0u=0 on M. By using the above result, we show that a non-negative biminimal properly immersed submanifold M in a complete Riemannian manifold N   with 0⩾KN⩾−L(1+distN(⋅,q0)2)α2 is minimal. These results give affirmative partial answers to the global version of generalized Chenʼs conjecture for biharmonic submanifolds.