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Navigating the Space of Compact CMC Hypersurfaces in Spheres, Part I

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arxiv 2503.13823 v1 pith:LRPLISBZ submitted 2025-03-18 math.DG

classification math.DG
keywords hypersurfacesembeddedexamplesspheresareacliffordcompactconjecture
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abstract

In this paper, we describe a family of embedded hypersurfaces with constant mean curvature (CMC) in the $(n+1)$-dimensional unit sphere. In the process, we provide evidence for new CMC embedded examples. In particular, for some examples with $H=0$, we verify Yau's conjecture stating that among the embedded, non-totally umbilical minimal hypersurfaces in spheres, the Clifford hypersurfaces have the least area.

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  1. New Explicit Eigenfunctions of the stability operator on some minimal hypersurfaces

    math.DG 2026-07 accept novelty 5.0 of 10

    For minimal immersions ϕ(y,z,t)=(f(t)y,f₂(t)z,f₁(t)), the functions ω(t)y_i z_j with ω=f^{-k}f₂^{-ℓ} are eigenfunctions of the stability operator for eigenvalue −n, implying index ≥ kℓ+3k+3ℓ+8.

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