Singular behavior and generic regularity of min-max minimal hypersurfaces

TL;DR

This paper investigates the regularity and singular behavior of min-max minimal hypersurfaces in 8-dimensional manifolds, establishing generic existence of smooth hypersurfaces under positive Ricci curvature and bounding singularities in a broader setting.

Contribution

It extends generic regularity results to min-max minimal hypersurfaces beyond area-minimizing cases, providing a universal estimate relating singular points and index in any dimension.

Findings

Existence of smooth minimal hypersurfaces in generic 8D manifolds with positive Ricci curvature.

Dense set of metrics with minimal hypersurfaces having at most one singular point.

Universal estimate linking singular points and index for min-max hypersurfaces in any dimension.

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