L~2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

摘要

Let x : M~m → M, m ≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply connected manifold ˉM with sectional curvature satisfying ?k~2 ≤ K _(ˉM) ≤ 0, for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L~m-norm. If K _M not identical with 0, assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L~2 harmonic 1-forms on M has finite dimension. Moreover, there exists a constant Λ>0, explicitly computed, such that if the total curvature is bounded from above by Λ then there are no nontrivial L~2-harmonic 1-forms on M.