A New Poincar E Type Inequality

Using the Green's function and some comparison theorems, we obtain a lower bound on the rst Dirichlet eigenvalue for a domain D on a complete manifold with curvature bounded from above. And the lower bound is given explicitly in terms of the diameter of D and the dimension of D. This result can be considered as an analogue for nonpositively curved manifolds of Li-Schoen L-Sc] and Li-Yau's L-Ya] theorems for nonnegatively curved manifolds. We also give conditions under which a minimal hypersurface is stable in spaces with constant curvature. 1 The Main Theorem The Poincar e inequality is one of the fundamental inequalities in the study of partial diierential equations. It is one of the essential tools for the derivation of many of the a priori estimates for the solutions of the diierential equations. In this paper, we will derive a Poincar e inequality for domains in nonpositively curved manifolds and in minimal submanifolds of a nonposi-tively curved manifold.