The structure of stable minimal hypersurfaces in I\!\!Rⁿ⁺¹

We provide a new topological obstruction for complete stable minimal hypersurfaces in IR n+1 . For n 3, we prove that a complete orientable stable minimal hypersurface in IR n+1 cannot have more than one end by showing the existence of a bounded harmonic function based on the Sobolev inequality for minimal submanifolds [MS] and by applying the Liouville theorem for harmonic functions due to Schoen-Yau [SY].