Estimating the Morse index of free boundary minimal hypersurfaces through covering arguments

TL;DR

This paper extends Song's method to free boundary minimal hypersurfaces in manifolds of dimension 3 to 7, demonstrating that their Morse index increases linearly with the sum of Betti numbers, influenced by boundary area.

Contribution

It adapts existing techniques to the free boundary setting, establishing a linear growth relation between Morse index and topological complexity.

Findings

Morse index grows linearly with Betti numbers.

The growth constant depends on the boundary area.

Method applicable to manifolds of dimension 3 to 7.

Abstract

Given a compact Riemannian manifold, of dimension between 3 and 7, with boundary, we adapt Song's method in Song (2023) to the free boundary case to show that the Morse index of a free boundary minimal hypersurface grows linearly with the sum of its Betti numbers, where the constant of growth depends on the area of the free boundary minimal hypersurface in question.