L2L2 harmonic forms on submanifolds in a Hadamard manifold

In the present paper, we first show that an n−1n-super-stable complete minimal submanifold in Rn+mRn+m admits no nontrivial L2L2 harmonic 11-forms, and has only one end, which generalized Cao–Shen–Zhu’s result on stable minimal hypersurface in Rn+1Rn+1. For L2L2 harmonic forms of higher order, we prove a vanishing and finiteness theorem under the assumptions on Schrödinger operators involving the squared norm of the traceless second fundamental form. Finally, if the potential function of the Schrödinger operator is a multiple of the squared norm of the second fundamental form, we can further show that the hypersurface has only one end or finitely many ends.