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Minimal hypersurfaces for generic metrics in dimension 8

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arxiv 2205.01047 v1 pith:4RPQI6I6 submitted 2022-05-02 math.DG math.AP

classification math.DGmath.AP
keywords genericminimaldimensionhypersurfaceseightmetricsalmgren-pittsclosed
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces

    math.DG 2026-07 accept novelty 8.0 of 10

    For d≥3 and c≥2, open sets of metrics force every mod-2 area-minimizer to have singular set of Hausdorff dimension at least d−3; the Veronese RP2 cone is mod-2 minimizing.

  2. Infinitely Many Surfaces with Prescribed Mean Curvature in the Presence of a Strictly Stable Minimal Surface

    math.DG 2025-02 conditional novelty 8.0 of 10

    On any closed 3-7 dimensional manifold that contains a strictly stable minimal surface, a large class of prescribed-mean-curvature functions admit infinitely many distinct almost embedded hypersurfaces.

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