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arXiv · 2602.03984

Equivariant min-max theory and the spherical Bernstein problem in $\mathbb{S}^4$

Tongrui Wang·Zhichao Wang·Xin Zhou
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Abstract

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 establish the regularity of solutions to the $G$-equivariant Plateau problem and the $G$-equivariant isotopy area minimization problem.

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Tongrui Wang, Zhichao Wang, Xin Zhou. 2026-02-03. Equivariant min-max theory and the spherical Bernstein problem in $\mathbb{S}^4$. https://arxiv.org/abs/2602.03984

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