On minimal hypersurfaces with finite harmonic indices

Abstract

We introduce the concepts of harmonic stability and harmonic index for a complete minimal hypersurface in $R^{n+1}(n\leq3)$ and prove that the hypersurface has only finitely many ends if its harmonic index is finite. Furthermore, the number of ends is bounded from above by 1 plus the harmonic index. Each end has a representation of nonnegative harmonic function, and these functions form a partition of unity. We also give an explicit estimate of the harmonic index for a class of special minimal hypersurfaces, namely, minimal hypersurfaces with finite total scalar curvature. It is shown that for such a submanifold the space of bounded harmonic functions is exactly generated by the representation functions of the ends.