STRUCTURE OF STABLE MINIMAL HYPERSURFACES IN A RIEMANNIAN MANIFOLD OF NONNEGATIVE RICCI CURVATURE

Abstract

Let N be a complete Riemannian manifold with nonnegative Ricci curvature and let M be a complete noncompact oriented stable minimal hypersurface in N. We prove that if M has at least two ends and <TEX>${\int}_M{\mid}A{\mid}^2\;dv={\infty}$</TEX>, then M admits a nonconstant harmonic function with finite Dirichlet integral, where A is the second fundamental form of M. We also show that the space of <TEX>$L^2$</TEX> harmonic 1-forms on such a stable minimal hypersurface is not trivial. Our result is a generalization of one of main results in [12] because if N has nonnegative sectional curvature, then M admits no nonconstant harmonic functions with finite Dirichlet integral. And our result recovers a main theorem in [3] as a corollary.