Complete submanifolds with bounded mean curvature in a Hadamard manifold

Let Q be an mm-dimensional Hadamard manifold and let MM be an nn-dimensional complete non-compact submanifold in Q. Let κκ be a non-positive constant and suppose that the (n−1)(n−1)-th Ricci curvature of Q is no less than (n−1)κ(n−1)κ. Denote by ΔΔ, AA and H the Laplacian operator, the second fundamental form and the mean curvature vector of MM, respectively. Assume that |H|≤H0<∞ and when κ=0κ=0 we also assume that |H|≥H1>0. In the present paper, we prove finiteness theorems on ends of MM using the theory of L2L2 harmonic 11-forms. In particular, we show that if n≥3n≥3 and if the index of the operator L≡Δ−(n−1)κ+n−12|A|2 is finite, then MM has finitely many ends; if n≥2n≥2 and if the index of LL is zero, then MM has only one end.