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

TL;DR

This paper constructs a new embedded non-equatorial minimal hypersphere in the 4-sphere, solving Chern's spherical Bernstein problem using equivariant min-max theory and confirming a long-standing conjecture.

Contribution

It introduces an equivariant min-max framework for G-invariant minimal hypersurfaces with reduced genus bounds, leading to new solutions in $S^4$ and confirming a 1986 conjecture.

Findings

Constructed a non-equatorial minimal hypersphere in $S^4$

Confirmed the assertion by Pitts-Rubinstein (1986)

Established regularity for G-equivariant Plateau and isotopy problems

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.