Index growth not imputable to topology

TL;DR

This paper establishes lower bounds on the Morse index of specific minimal hypersurfaces in four-dimensional spheres and balls, revealing linear index growth phenomena that contrast with three-dimensional cases.

Contribution

It introduces a partitioning method for analyzing Morse index in higher dimensions, extending previous approaches to general compact Lie group actions.

Findings

Linear index growth for sequences of fixed topological type

Effective lower bounds on Morse index for minimal hypersurfaces

Contrast with three-dimensional minimal hypersurface behavior

Abstract

We employ partitioning methods, in the spirit of Montiel--Ros but here recast for general actions of compact Lie groups, to prove effective lower bounds on the Morse index of certain families of closed minimal hypersurfaces in the round four-dimensional sphere, and of free boundary minimal hypersurfaces in the Euclidean four-dimensional ball. Our analysis reveals, in particular, phenomena of linear index growth for sequences of minimal hypersurfaces of fixed topological type, in strong contrast to the three-dimensional scenario.