Min-max hypersurface in manifold of positive Ricci curvature

Xin Zhou Massachusetts Institute of Technology

Differential Geometry mathscidoc:1610.10031

Distinguished Paper Award in 2020

2015
In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the Morse index, area and multiplicity of this singular min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and has Morse index one, or is a double cover of a non-orientable stable minimal hypersurface. As an essential technical tool, we prove a stronger version of the discretization theorem. The discretization theorem, first developed by Marques-Neves in their proof of the Willmore conjecture \cite{MN12}, is a bridge to connect sweepouts appearing naturally in geometry to sweepouts used in the min-max theory. Our result removes a critical assumption of \cite{MN12}, called the no mass concentration condition, and hence confirms a conjecture by Marques-Neves in \cite{MN12}.
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@inproceedings{xin2015min-max,
  title={Min-max hypersurface in manifold of positive Ricci curvature},
  author={Xin Zhou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161010161455674587110},
  year={2015},
}
Xin Zhou. Min-max hypersurface in manifold of positive Ricci curvature. 2015. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161010161455674587110.
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