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

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.

arXiv comment31 pages, 3 figures. First version contained a mistake pointed out by Antoine Song. Final version accepted in J. Reine Angew