Minimal hypersurfaces for generic metrics in dimension 8

Abstract We show that in an 8-dimensional closed Riemmanian manifold with $C^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> -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. En route to our main results, we have proved a sheeting theorem for minimal hypersurfaces in dimension 8 (Appendix C), which gives an affirmed answer to a question asked by Ilmanen in dimension 8.