Equivariant min-max theory and the spherical Bernstein problem in $\mathbb{S}^4$
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| Format: | Preprint |
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2026
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| _version_ | 1866917246954110976 |
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| author | Wang, Tongrui Wang, Zhichao Zhou, Xin |
| author_facet | Wang, Tongrui Wang, Zhichao Zhou, Xin |
| contents | We construct an embedded non-equatorial minimal hypersphere in the unit $4$-sphere $\mathbb{S}^4$, which provides a new resolution of Chern's spherical Bernstein problem in $\mathbb{S}^4$. The construction is based on our equivariant min-max theory for $G$-invariant minimal hypersurfaces with reduced genus bound, where $G$ is a compact Lie group acting by isometries on a closed Riemannian manifold with $3$-dimensional orbit space. This confirms an assertion made by Pitts-Rubinstein in 1986. We also show the regularity for the solutions of the $G$-equivariant Plateau problem and the $G$-equivariant isotopy area minimization problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03984 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivariant min-max theory and the spherical Bernstein problem in $\mathbb{S}^4$ Wang, Tongrui Wang, Zhichao Zhou, Xin Differential Geometry 53A10, 53C42 We construct an embedded non-equatorial minimal hypersphere in the unit $4$-sphere $\mathbb{S}^4$, which provides a new resolution of Chern's spherical Bernstein problem in $\mathbb{S}^4$. The construction is based on our equivariant min-max theory for $G$-invariant minimal hypersurfaces with reduced genus bound, where $G$ is a compact Lie group acting by isometries on a closed Riemannian manifold with $3$-dimensional orbit space. This confirms an assertion made by Pitts-Rubinstein in 1986. We also show the regularity for the solutions of the $G$-equivariant Plateau problem and the $G$-equivariant isotopy area minimization problem. |
| title | Equivariant min-max theory and the spherical Bernstein problem in $\mathbb{S}^4$ |
| topic | Differential Geometry 53A10, 53C42 |
| url | https://arxiv.org/abs/2602.03984 |