L2 HARMONIC 1-FORMS ON COMPLETE SUBMANIFOLDS IN EUCLIDEAN SPACE

Let M-n (n >= 3) be an n-dimensional complete noncompact oriented submanifold in an (n + p)-dimensional Euclidean space Rn+p with finite total mean curvature, i.e, integral(M) vertical bar H vertical bar(n) < infinity, where H is the mean curvature vector of M. Then we prove that each end of M must be non-parabolic. Denote by phi the traceless second fundamental form of M. We also prove that if integral(M) vertical bar phi vertical bar(n) < C(n), where C(n) is an an explicit positive constant, then there are no nontrivial L-2 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.