The structure of weakly stable minimal hypersurfaces

In this short communication, we announce results from our research on the structure of complete noncompact oriented weakly stable minimal hypersurfaces in a manifold of nonnegative sectional curvature. In particular, a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, must have only one end; any complete noncompact oriented weakly stable minimal hypersurface has only one end if the complete oriented ambient manifold Nm, m > or = 7, has nonnegative sectional curvature and Ricci curvature bounded below by a positive constant; a complete oriented weakly stable minimal hypersurface in Rm, m > or = 4, with finite total scalar curvature is a hyperplane.