<p>We show that in an 8-dimensional closed Riemmanian manifold with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="222_2025_1333_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi mathvariant="normal">∞</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$C^{\infty }$</EquationSource> </InlineEquation>-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&#xa0;<InternalRef RefID="Sec27">C</InternalRef>), which gives an affirmed answer to a question asked by Ilmanen in dimension 8.</p>

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Minimal hypersurfaces for generic metrics in dimension 8

  • Yangyang Li,
  • Zhihan Wang

摘要

We show that in an 8-dimensional closed Riemmanian manifold with C $C^{\infty }$ -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. 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.