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

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.

78 pages, 6 figures. Added a construction of an embedded minimal S^1 \times S^2 in S^4; minor revisions and typo corrections. Comments are welcome!