The topological structure of complete noncompact submanifolds in spheres

In this paper, we first show that δ-super stable complete noncompact minimal submanifolds in Sm+n or Rm+n with δ>(m−1m)2 admit no nontrivial L2 harmonic 1-forms and have only one nonparabolic end, which generalizes Cao-Shen-Zhu's result in [2] on stable minimal hypersurface in Rm+1 and Lin's result in [13] on m−1m-super stable minimal submanifolds in Rm+n. Second, we prove that the dimension of the space of L2 harmonic p-forms on Mm is zero or finite and there is only one nonparabolic end or finitely many nonparabolic ends of M under the assumptions on the Schrödinger operators involving the squared norm of the traceless second fundamental form.