Existence of infinitely many minimal hypersurfaces in higher-dimensional closed manifolds with generic metrics

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 the equidistribution of the minimal hypersurfaces realizing min-max widths conjectured by F.C. Marques, A. Neves, and A. Song in [19].