Morse index of multiplicity one min-max minimal hypersurfaces

TL;DR

This paper establishes that, generically, the Morse index of certain minimal hypersurfaces obtained via min-max methods equals the dimension of the homology class they represent, advancing Morse theory for the area functional.

Contribution

It proves the Morse index of multiplicity one min-max minimal hypersurfaces matches the homology class dimension, confirming part of a broader Morse theory program.

Findings

Morse index equals homology class dimension for generic minimal hypersurfaces

Supports development of Morse theory for the area functional

Advances understanding of min-max minimal hypersurfaces

Abstract

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-index}, \cite{neves-icm}) proposed by the authors with the goal of developing a Morse theory for the area functional.