Catenoid-sharp index-topology estimates for minimal hypersurfaces

Abstract: We show that for an embedded two-sided minimal hypersurface in $\mathbb{R}<sup>N$, 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 $\operatorname{Spin}(7)$ analogue of Theorem 10.1 of Chodosh--Gianocca: a complete two-sided minimal immersion $M<sup>7\to\mathbb{R}<sup>8$ of index one and finite total curvature is a higher-dimensional catenoid.