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On $\delta$-Stable Minimal Hypersurfaces in $\mathbb{R}^{n+1}$

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arxiv 2407.03222 v1 pith:NGSO2AGI submitted 2024-07-03 math.DG math.AP

Han Hong , Haizhong Li , Gaoming Wang This is my paper
classification math.DGmath.AP
keywords deltastablehypersurfacesminimalmathbbfracbernsteincatenoid
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abstract

In this paper, we extend several results established for stable minimal hypersurfaces to $\delta$-stable minimal hypersurfaces. These include the regularity and compactness theorems for immersed $\delta$-stable minimal hypersurfaces in $\mathbb{R}^{n+1}$ when $n \geq 3$ and $\delta > \frac{n-2}{n}$, as well as the $\delta$-stable Bernstein theorem for $n=3$ and $n=4$ for properly immersion. The range of $\delta$ is optimal, as the $n$-dimensional catenoid in $\mathbb{R}^{n+1}$ is $\frac{n-2}{n}$-stable.

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Cited by 3 Pith papers

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

  1. Regularity of branched stable minimal immersed hypersurfaces

    math.DG 2026-07 conditional novelty 8.0 of 10

    For stable minimal immersed hypersurfaces with finite (n−2)-dimensional singular set, the non-branch singular set has dimension at most n−3, and examples attain this bound in every dimension n≥3.

  2. Minimal hypersurfaces of Morse index one

    math.DG 2026-07 accept novelty 7.0 of 10

    Any complete connected embedded minimal hypersurface in R^{n+1} with finite total curvature and Morse index one is a higher-dimensional catenoid.

  3. Complete two-sided $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$

    math.DG 2025-07 conditional novelty 6.0 of 10

    For n=3,4,5 and δ above thresholds δ0(n), complete two-sided δ-stable minimal hypersurfaces in R^{n+1} have Euclidean volume growth, and for δ above δ1(n) they are hyperplanes.

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