Finiteness of index and total scalar curvature for minimal hypersurfaces

Abstract:

Let ${M^n},n \geq 3$, be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for $n = 3,4,5,{\text { or }}6$, the index of ${M^n}$ is finite if and only if the total scalar curvature of ${M^n}$ is finite, provided that the volume growth of ${M^n}$ is bounded by a constant times ${r^n}$, where $r$ is the Euclidean distance function. We also note that this result does not hold for $n \geq 8$. Moreover, we show that the index of ${M^n}$ is bounded by a constant multiple of the total scalar curvature for all $n \geq 3$, without any assumptions on the volume growth of ${M^n}$.