Singular behavior and generic regularity of min-max minimal hypersurfaces

Abstract

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): $${\mathcal{H}}^0({\mathcal{S}}_{nm}(\Sigma ))+\mathrm{Index}(\Sigma )\le 1$$ where ${\mathcal{S}}_{nm}(\Sigma )$ denotes the set of singular points of $\Sigma$ with a unique tangent cone non-area minimizing on either side.

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