Catenoid-sharp index-topology estimates for minimal hypersurfaces

We show that for an embedded two-sided minimal hypersurface in ${\mathbb{R}}^N$, there is a lower bound for the index in terms of the first Betti number and the number of ends, using ideas from the recent work of Chodosh–Gianocca. This estimate is sharp for the higher-dimensional catenoid. We also obtain a $\mathrm{Spin}(7)$ analogue of Theorem 10.1 of Chodosh–Gianocca: a complete two-sided minimal immersion $M^7\to {\mathbb{R}}^8$ of index one and finite total curvature is a higher-dimensional catenoid.