L 2 Harmonic 1-Forms on Minimal Submanifolds in Spheres

We study a complete noncompact minimal submanifold M ⁿ in a sphere S ^n+p . We prove there is no nontrivial L ² harmonic 1-form and at most one nonparabolic end on M if the total curvature is bounded from above by a constant depending only on n. The rigidity theorem is a generalized version of Ni’s, Yun’s and the second author’s results on submanifolds in Euclidean spaces and Seo’s result on minimal submanifolds in hyperbolic spaces.