# Existence of infinitely many minimal hypersurfaces in low dimensions, after F.C. Marques, A.A. Neves et A. Song (Bourbaki Seminar)

## TL;DR

This paper discusses the proof that in dimensions 3 to 7, any smooth closed Riemannian manifold contains infinitely many closed embedded minimal surfaces, extending classical results about geodesics in 3D.

## Contribution

It presents the recent proof by Antoine Song that generalizes Morse's classical result to higher dimensions, showing the existence of infinitely many minimal hypersurfaces.

## Key findings

- Existence of infinitely many minimal hypersurfaces in dimensions 3 to 7.
- Extension of Morse's geodesic result to higher dimensions.
- Proof techniques introduced by Antoine Song.

## Abstract

A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher dimensions and, more precisely, on any smooth closed riemannian manifod of arbitrary dimension between 3 and 7 there exists infinitely many closed embedded minimal surfaces. We are going to present the origins of this theorem as well as it's proof given recently by Antoine Song.