Minimal hypersurfaces for generic metrics in dimension 8

TL;DR

This paper proves that in 8-dimensional closed Riemannian manifolds with generic metrics, all minimal hypersurfaces are smooth and nondegenerate, confirming a key regularity conjecture and extending known properties to this dimension.

Contribution

It establishes the full generic regularity of minimal hypersurfaces in dimension eight and generalizes several geometric properties previously known only in lower dimensions.

Findings

All minimal hypersurfaces are smooth and nondegenerate in generic metrics in dimension 8.

Confirms the full regularity conjecture for minimal hypersurfaces in dimension 8.

Extends generic geometric properties of min-max minimal hypersurfaces to 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.