Gap theorems on hypersurfaces in spheres

We study a complete noncompact hypersurface MnMn isometrically immersed in an (n+1)(n+1)-dimensional sphere Sn+1Sn+1(n≥3)(n≥3). We prove that there is no non-trivial L2L2-harmonic 2-form on M, if the length of the second fundamental form is less than a fixed constant. We also showed that the same conclusion holds if the scale-invariant total tracefree curvature is bounded above by a small constant depending only on n  . These results are generalized versions of the result of Cheng and Zhou on bounded harmonic functions with finite Dirichlet integral and the one of Fang and the author on L2L2 harmonic 1-forms.