On minimal hypersurfaces with finite harmonic indices

We introduce the concepts of harmonic stability and harmonic index for a co mplete minimal hypersurface in Rn+1 (n greater than or equal to 3) and prov e that the hypersurface has only finitely many ends if its harmonic index i s finite. Furthermore, the number of ends is bounded from above by I plus t he harmonic index. Each end has a representation of nonnegative harmonic fu nction, and these,functions form a partition of unity. We also give an expl icit estimate of the harmonic index for a class of special minimal hypersur faces, namely, minimal hypersurfaces with finite total scalar curvature. It is shown that for such a submanifold the space of bounded harmonic functio ns is exactly generated by the representation functions of the ends.