# Existence of Infinitely Many Minimal Hypersurfaces in Higher-dimensional Closed Manifolds with Generic Metrics

## TL;DR

This paper proves that generic metrics on high-dimensional closed manifolds guarantee infinitely many singular minimal hypersurfaces, and for lower dimensions, minimal hypersurfaces are dense in the space of metrics, supporting conjectures on their distribution.

## Contribution

It establishes the existence of infinitely many minimal hypersurfaces in high dimensions and density results in lower dimensions for generic metrics, advancing understanding of minimal hypersurface distribution.

## Key findings

- Infinitely many singular minimal hypersurfaces in high dimensions with generic metrics.
- Density of minimal hypersurfaces realizing min-max widths in dimensions 2 to 6.
- Partial support for the conjecture on equidistribution of minimal hypersurfaces.

## Abstract

In this paper, we show that a closed manifold $M^{n+1} (n \geq 7)$ endowed with a $C^\infty$-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal regularity. Moreover, for $2 \leq n \leq 6$, our argument also implies the denseness of the minimal hypersurfaces realizing min-max widths for generic metrics. This partially supports equidistribution of the minimal hypersurfaces realizing min-max widths conjectured by F.C. Marques, A. Neves and A. Song.