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Riemannian metrics on the sphere with Zoll families of minimal hypersurfaces

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arxiv 2112.01448 v1 pith:2NJCOC6A submitted 2021-12-02 math.DG math.AP

Lucas Ambrozio , Fernando C. Marques , André Neves This is my paper · ORCID
classification math.DGmath.AP
keywords minimalhypersurfacesmetricsspherecasefamiliesriemanniansmooth
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In this paper we construct smooth Riemannian metrics on the sphere which admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse Function Theorem in the tame maps setting of Hamilton. This answers a question of Yau on perturbations of minimal hypersurfaces in positive Ricci curvature. We also consider the case of the projective space and characterize those metrics on the sphere with minimal equators.

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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. Volume spectrum of fiber bundles and the widths of Berger spheres

    math.DG 2025-05 conditional novelty 7.0 of 10

    Fiber bundles inherit an upper bound for their min-max widths from their base, and this yields exact low widths for Berger spheres and sphere products.

  2. The p-widths of $RP^2$

    math.DG 2025-01 conditional novelty 6.0 of 10

    The p-widths of (RP^2, g_std) are ω_p = 2π floor((1+sqrt(1+8p))/4), achieved by Z2-invariant polynomial sweepouts.

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