Vanishing and Topological Sphere Theorems for Submanifolds in a Hyperbolic Space

We extend the vanishing and sphere theorems due to Lawson, Simons, Xin, Shiohama and Xu. By using the techniques of calculus of variations in the geometric measure theory, we prove the vanishing theorem for homology groups of submanifolds in the hyperbolic space H n (c) with negative constant curvature c. Moreover, we obtain a topological sphere theorem for certain complete submanifolds in H n (c).