Morse index and multiplicity of min-max minimal hypersurfaces

TL;DR

This paper advances min-max theory for minimal hypersurfaces by providing the first general Morse index bounds and resolving the multiplicity problem for one-parameter sweepouts, significantly enhancing understanding of minimal hypersurface properties.

Contribution

It introduces the first general Morse index bounds for min-max minimal hypersurfaces and solves the multiplicity problem in the classical one-parameter case.

Findings

Established Morse index bounds for min-max minimal hypersurfaces.

Resolved the multiplicity problem for one-parameter sweepouts.

Enhanced the theoretical framework of min-max minimal hypersurface theory.

Abstract

The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the min-max minimal hypersurface. We advance the theory further and prove the first general Morse index bounds for minimal hypersurfaces produced by it. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.