Singular behavior and generic regularity of min-max minimal hypersurfaces

We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set of 8-dimensional Riemannian metrics there exists a minimal hypersurface with at most one singular point. This extends previous work on generic regularity that only dealt with area-minimizing hypersurfaces. These results are a consequence of a more general estimate for a one-parameter min-max minimal hypersurface $\Sigma \subset (M,g)$ (valid in any dimension): $$\mathcal H^{0} (\mathcal{S}_{nm}(\Sigma)) +{\rm Index}(\Sigma) \leq 1$$ where $\mathcal{S}_{nm}(\Sigma)$ denotes the set of singular points of $\Sigma$ with a unique tangent cone non-area minimizing on either side.