Total curvature and L 2 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., ${\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}$ , in an (n + p)-dimensional simply connected space form N ^n+p (c) of constant curvature c, where |H| and |A|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 ${\int_M|H|^n < \infty}$ ; (ii) n ≥ 5, c = ?1 and ${|H| < 1-\frac{2}{\sqrt n}}$ ; (iii) n ≥ 3, c = 1 and |H| 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.