Strong multiplicity one theorems and homological min-max theory

TL;DR

This work advances Almgren–Pitts min-max theory by developing restrictive, mass-bounded variants of both homotopic and homological min-max frameworks, together with annular-replacement and almost-minimizing techniques, to guarantee multiplicity-one minimal hypersurfaces in the critical sets. By combining a detailed deformation theory (pull-tight with mass bounds and DEF/PT constructions) with careful metric perturbations and cobordism arguments, the authors prove that certain high-index min-max widths of the unit 3-sphere satisfy strict bounds, notably showing \\omega_{13}(S^3) < 8\\pi and placing widths \\omega_{10} - \\omega_{13} between \\omega_{13} \\nobreak= 2\\pi^2 and 8\\pi. The resulting strong multiplicity-one theorems (two variants) ensure the relevant minimizers have multiplicity-one, two-sided minimal hypersurfaces, with a variety of applications to constructing infinitely many minimal hypersurfaces and to Yau-type conjectures in 3-manifolds. The methods combine precise interpolation, mass-control deformations, and a robust variational framework, yielding new rigidity and existence results in high-dimensional geometric analysis.

Abstract

It was asked by Marques-Neves which min-max $p$-widths of the unit $3$-sphere lie strictly between $2π^2$ and $8π$. We show that the 10th to the 13th widths do. More generally, we prove stronger versions of X. Zhou's multiplicity one theorem.

Figures (4)

• Figure 1:
• Figure 2: Construction of $W"_i=\mathrm{dmn}(\Psi"^\omega_i)$.
• Figure 3: Construction of $\hat{W}_i=\mathrm{dmn}(\hat{\Psi}_i)$.
• Figure 4: The final parameter space $\hat{\hat{W}}^\omega+\partial A.$

Theorems & Definitions (88)

• Theorem 1.1
• Theorem 1.2: Strong multiplicity one theorem I
• Theorem 1.3: Strong multiplicity one theorem II
• Remark 1.4
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Proposition 2.6: MN17