Comparing the Morse index and the first Betti number of minimal hypersurfaces

TL;DR

This paper establishes a linear lower bound on the Morse index of closed minimal hypersurfaces in certain positively curved manifolds based on their first Betti number, confirming a broad conjecture.

Contribution

It extends previous methods to prove a general relation between Morse index and topology for minimal hypersurfaces in various ambient spaces.

Findings

Morse index is bounded below by a linear function of the first Betti number.

The relation holds in all compact rank one symmetric spaces.

The results confirm the Schoen and Marques-Neves conjecture in multiple settings.

Abstract

By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due to Schoen and Marques-Neves for a wide class of ambient spaces.