[Paper Review] Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\leq n\leq 6$

Xin Zhou|arXiv (Cornell University)|Oct 7, 2012

Geometric Analysis and Curvature Flows13 references9 citations

TL;DR

This paper characterizes the Morse index, area, and multiplicity of the min-max minimal hypersurface in a closed, orientable Riemannian manifold $(M^{n+1}, g)$ with positive Ricci curvature and $2 \leq n \leq 6$. Using Almgren-Pitts min-max theory and a novel sweepout construction, it proves the hypersurface is either an orientable, index-one, multiplicity-one minimal surface of least area, or a non-orientable double cover of a least-area minimal hypersurface.

ABSTRACT

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts in \cite{A2}\cite{P} 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.

Motivation & Objective

• . The paper aims to determine the geometric and topological properties of the min-max minimal hypersurface constructed via Almgren-Pitts theory in manifolds with positive Ricci curvature.
• It seeks to resolve the long-standing conjecture that such min-max hypersurfaces should have Morse index at most one.
• The study focuses on characterizing the Morse index, volume (area), and multiplicity of the min-max hypersurface corresponding to the fundamental class.
• It investigates the dichotomy between orientable and non-orientable minimal hypersurfaces in this setting.
• The objective includes proving that the least area minimal hypersurface exists and is realized via min-max, even without a priori Morse index bounds.

Proposed method

• . The authors embed any closed embedded minimal hypersurface into a 'good' sweepout, a continuous family of hypersurfaces that captures the topology of the manifold.
• They discretize these sweepouts to fit the discrete setting of Almgren-Pitts min-max theory.
• A key technique is constructing a Morse function on the manifold that agrees with a level-set foliation near the hypersurface, ensuring the sweepout lies in the same homotopy class.
• The construction relies on perturbing a distance function to produce a Morse function, which allows defining a sweepout via level sets.
• The method distinguishes between orientable and non-orientable cases, particularly in handling the fundamental class and homotopy class in the space of integral currents.
• The proof uses the fact that in manifolds with $\mathrm{Ric}_g > 0$, no stable minimal hypersurfaces exist, which forces the min-max hypersurface to be unstable and thus of index at least one.

Experimental results

Research questions

• RQ1 . What is the Morse index of the min-max minimal hypersurface corresponding to the fundamental class in a manifold with positive Ricci curvature and $2 \leq n \leq 6$?
• RQ2 . Does the min-max hypersurface achieve the least area among all closed embedded minimal hypersurfaces in such manifolds?
• RQ3 . Under what conditions is the min-max hypersurface orientable versus non-orientable?
• RQ4 . Can the multiplicity of the min-max hypersurface be characterized in terms of the topology of the ambient manifold?
• RQ5 . Is the existence of a least-area minimal hypersurface in such manifolds a consequence of min-max theory rather than compactness of stable sequences?

Key findings

• . The min-max hypersurface is either orientable with Morse index one and multiplicity one, or non-orientable with multiplicity two.
• . In the orientable case, the area of the hypersurface equals $W_M$, the infimum of the volume over all minimal hypersurfaces.
• . In the non-orientable case, the area of the hypersurface satisfies $2V(\Sigma) = W_M$, meaning it achieves the least area among all closed embedded minimal hypersurfaces.
• . The least area minimal hypersurface exists and is realized via the min-max construction, even though the class of all minimal hypersurfaces lacks a priori Morse index bounds.
• . The result confirms that in $\mathrm{Ric}_g > 0$ manifolds, the min-max hypersurface is either index one (orientable) or a double cover of a least-area non-orientable minimal hypersurface.
• . The construction shows that all sweepouts from minimal hypersurfaces lie in the same homotopy class in $\pi_1(\mathrm{Z}_n(M), \{0\})$, enabling the application of Almgren-Pitts theory.