Min-max hypersurface in manifold of positive Ricci curvature
By Xin Zhou. 2015
54 citations recorded by the source corpus.
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}.
Subfields: Applied Mathematics, Computational Mechanics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/36c95a531dceb97a2a3204413467b9f71a0921c5
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Computational Mechanics
In Geometry and Topology
Citations in the corpus
- Morse index of multiplicity one min-max minimal hypersurfaces (2015, background): “The one-parameter case in manifolds with positive Ricci curvature had been studied before by Zhou [35, 36], who did important work extending to high dimensions previous work of the authors [19].”
- The Allen–Cahn equation on closed manifolds (2016, methodology): “We apply a result from [65] in a discrete setting.”
- Min-max for phase transitions and the existence of embedded minimal hypersurfaces (2015, methodology): “Concerning the general case, in a recent work [44], Xin Zhou showed that in fact it is not necessary to assume the no concentration of mass condition.”
- On the Multiplicity One Conjecture in min-max theory (2019, unknown): “This conjecture was proved earlier for 1-parameter min-max constructions under positive Ricci curvature assumption by Marques-Neves [25], the author [48, 49], and Ketover-Marques-Neves [22].”