A new Poincaré type inequality

Using the Green's function and some comparison theorems, we obtain a lower bound on the first 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 differential equations. It is one of the essential tools for the derivation of many of the a priori estimates for the solutions of the differential equations. In this paper, we will derive a Poincar'e inequality for domains in nonpositively curved manifolds and in minimal submanifolds of a nonpositively curved manifold. 1 1991 Mathemati..