L2 harmonic 1-forms on complete submanifolds in Euclidean space

Abstract

Let Mⁿ (n ≥ 3) be an n-dimensional complete noncompact oriented submanifold in an (n+p)-dimensional Euclidean space R^n+p with finite total mean curvature, i.e, ∫_M|H|ⁿ < ∞, where H is the mean curvature vector of M. Then we prove that each end of M must be non-parabolic. Denote by φ the traceless second fundamental form of M. We also prove that if ∫_M|φ|ⁿ < C (n), where C (n) is an an explicit positive constant, then there are no nontrivial L² harmonic 1-forms on M and the first de Rham's cohomology group with compact support of M is trivial. As corollaries, such a submanifold has only one end. This implies that such a minimal submanifold is plane.