Total curvature and L2 harmonic 1-forms on complete submanifolds in space forms

Let M-n be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., integral(M)(vertical bar A vertical bar(2) - n vertical bar H vertical bar(2))(n/2) < infinity, in an (n + p)-dimensional simply connected space form Nn+p (c) of constant curvature c, where vertical bar H vertical bar and vertical bar A vertical bar(2) are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n >= 3, c = 0 and integral(M) vertical bar H vertical bar(n) < infinity; (ii) n >= 5, c = -1 and vertical bar H vertical bar < 1 - 2/root n; (iii) n >= 3, c = 1 and vertical bar H vertical bar is bounded, then the dimension of the space of L-2 harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.