A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold M n in a sphere S n + p . We prove that there admit no nontrivial L 2 -harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carronʼs, Yunʼs, Cavalcanteʼs and the first authorʼs results on submanifolds in Euclidean spaces and Seoʼs result on submanifolds in hyperbolic space without the condition of minimality.