Schoen–Marques–Neves index–topology conjecture for minimal hypersurfaces

Let $(N^{n+1},g)$ be a closed Riemannian manifold with positive Ricci curvature. A closed embedded minimal hypersurface is a smooth hypersurface $M^n\subset N^{n+1}$ with zero mean curvature; its Morse index is denoted by $\operatorname{index}(M)$, and $b_1(M)$ is its first Betti number. Schoen–Marques–Neves conjecture. There exists a positive constant $C$ such that, for every closed embedded minimal hypersurface $M^n\subset N^{n+1}$,

$$\operatorname{index}(M)\ge C\\, b_1(M).$$

This conjecture predicts that the Morse index controls the basic topology of minimal hypersurfaces in positively Ricci-curved manifolds. The stated claim was proved by Savo, building on earlier work of Ros.