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

Tongrui Wang, Zhichao Wang, Xin Zhou

TL;DR

This work develops a comprehensive equivariant min-max framework for minimal hypersurfaces in closed manifolds with a 3-dimensional orbit space, establishing genus-controlled existence and regularity results for $G$-invariant hypersurfaces. A central contribution is the equivariant min-max theorem, which produces $G$-invariant minimal hypersurfaces with explicit topological bounds via a genus inequality on the quotient surfaces. The theory is then applied to the spherical Bernstein problem in $S^4$, producing an embedded non-equatorial minimal hypersphere and, byproduct, an embedded minimal hypertorus, using an extended symmetry group to control the orbit space and exploit multiplicity-one arguments on the quotient. The results hinge on a robust regularity theory for equivariant Plateau problems and isotopy minimization, together with a replacement scheme and genus bounds that mirror the classical Simon–Smith–Meeks–Yau program but in the presence of group actions and orbit-space singularities. Overall, the paper provides a versatile, symmetry-driven variational approach to constructing and classifying highly symmetric minimal hypersurfaces, with potential implications for Yau-type conjectures and related geometric-analytic problems in higher dimensions.

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 show the regularity for the solutions of the $G$-equivariant Plateau problem and the $G$-equivariant isotopy area minimization problem.

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

TL;DR

This work develops a comprehensive equivariant min-max framework for minimal hypersurfaces in closed manifolds with a 3-dimensional orbit space, establishing genus-controlled existence and regularity results for -invariant hypersurfaces. A central contribution is the equivariant min-max theorem, which produces -invariant minimal hypersurfaces with explicit topological bounds via a genus inequality on the quotient surfaces. The theory is then applied to the spherical Bernstein problem in , producing an embedded non-equatorial minimal hypersphere and, byproduct, an embedded minimal hypertorus, using an extended symmetry group to control the orbit space and exploit multiplicity-one arguments on the quotient. The results hinge on a robust regularity theory for equivariant Plateau problems and isotopy minimization, together with a replacement scheme and genus bounds that mirror the classical Simon–Smith–Meeks–Yau program but in the presence of group actions and orbit-space singularities. Overall, the paper provides a versatile, symmetry-driven variational approach to constructing and classifying highly symmetric minimal hypersurfaces, with potential implications for Yau-type conjectures and related geometric-analytic problems in higher dimensions.

Abstract

We construct an embedded non-equatorial minimal hypersphere in the unit -sphere , which provides a new resolution of Chern's spherical Bernstein problem in . The construction is based on our equivariant min-max theory for -invariant minimal hypersurfaces with reduced genus bound, where is a compact Lie group acting by isometries on a closed Riemannian manifold with -dimensional orbit space. This confirms an assertion made by Pitts-Rubinstein in 1986. We also show the regularity for the solutions of the -equivariant Plateau problem and the -equivariant isotopy area minimization problem.
Paper Structure (27 sections, 48 theorems, 214 equations, 2 figures)

This paper contains 27 sections, 48 theorems, 214 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Equivariant min-max theory
  3. Equivariant isotopy area minimizing
  4. Outline
  5. Preliminary
  6. Group actions
  7. Local structure of $M/G$
  8. Local structure of $\Sigma/G$ and notations for $G$-hypersurfaces
  9. Equivariant reduction of area
  10. Geometric measure theory
  11. Equivariant $\gamma$-reduction
  12. Interior regularity of equivariant Plateau's problem
  13. Regularity near $p\in M\setminus M^{prin}$ with $\dim(P(p))=2$
  14. Regularity near $p\in M\setminus M^{prin}$ with $\dim(P(p))=1$
  15. Proof of Theorem \ref{['Thm: plateau problem']}
  16. ...and 12 more sections

Key Result

Theorem 1.2

Let $S^1$ act on $\mathbb S^4=\{(x,z_1,z_2)\in \mathbb R\times\mathbb C^2 : |x|^2+|z_1|^2+|z_2|^2=1 \}$ by the suspension of the Hopf action, i.e. $e^{i\theta}\cdot (x,z_1,z_2):=(x, e^{i\theta}z_1, e^{i\theta}z_2)$ for all $e^{i\theta}\in S^1$ and $(x,z_1,z_2)\in \mathbb S^4$. Then there exist an em

Figures (2)

  • Figure I:
  • Figure II:

Theorems & Definitions (133)

  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Equivariant Min-max Theory
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • Definition 2.1
  • ...and 123 more