An Improved Morse Index Bound of Min-Max Minimal Hypersurfaces

TL;DR

This paper improves the Morse index bounds for min-max minimal hypersurfaces in closed Riemannian manifolds, extending previous results and introducing new techniques that do not depend on bumpy metrics.

Contribution

It provides a generalized Morse index bound for minimal hypersurfaces using hierarchical deformations and restrictive min-max theory, applicable in broader settings.

Findings

Enhanced Morse index bounds for minimal hypersurfaces.

Techniques do not require bumpy metrics, broadening applicability.

Generalization of previous results to higher dimensions.

Abstract

In this paper, we give an improved Morse index bound of minimal hypersurfaces from Almgren-Pitts min-max construction in any closed Riemannian manifold $M^{n+1}$ $(n+1 \geq 3$), which generalizes a result by X. Zhou \cite{zhou_multiplicity_2019} for $3 \leq n+1 \leq 7$. The novel techniques are the construction of hierarchical deformations and the restrictive min-max theory. These techniques do not rely on bumpy metrics, and thus could be adapted to many other min-max settings.