## Journal of Differential Geometry

### Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric \geq 0$ and $2 \leq n \leq 6$

Xin Zhou

#### Abstract

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in F. Almgren, The theory of varifolds, and J. Pitts, Existence and regularity of minimal surfaces on Riemannian manifold, corresponding to the fundamental class of a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature with $2 \leq n \leq 6$. We characterize the Morse index, area, and multiplicity of this min-max hypersurface. In particular, we show that the min-max hypersurface is either orientable and of index one, or is a double cover of a non-orientable minimal hypersurface with least area among all closed embedded minimal hypersurfaces.

#### Article information

Source
J. Differential Geom., Volume 100, Number 1 (2015), 129-160.

Dates
First available in Project Euclid: 24 March 2015

https://projecteuclid.org/euclid.jdg/1427202766

Digital Object Identifier
doi:10.4310/jdg/1427202766

Mathematical Reviews number (MathSciNet)
MR3326576

Zentralblatt MATH identifier
1331.53092

#### Citation

Zhou, Xin. Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric \geq 0$ and $2 \leq n \leq 6$. J. Differential Geom. 100 (2015), no. 1, 129--160. doi:10.4310/jdg/1427202766. https://projecteuclid.org/euclid.jdg/1427202766