Minimal hypersurfaces for generic metrics in dimension 8

Abstract

We show that in an $8$-dimensional closed Riemmanian manifold with $C^{\infty }$-generic metrics, every minimal hypersurface is smooth and nondegenerate. This confirms a full generic regularity conjecture of minimal hypersurfaces in dimension eight. This also enables us to generalize many generic geometric properties of (Almgren-Pitts) min-max minimal hypersurfaces, previously only known in low dimensions, to dimension eight.

73 pages; Comments welcome