Minimal hyperspheres of arbitrarily large Morse index

TL;DR

This paper demonstrates that in four-dimensional Riemannian manifolds, minimal hyperspheres can have arbitrarily large Morse index regardless of their volume and topology, contrasting with known three-dimensional results.

Contribution

It introduces a method to construct metrics on S^4 with minimal hyperspheres of bounded volume but unbounded Morse index, revealing new phenomena in higher dimensions.

Findings

Constructed metrics on S^4 with minimal hyperspheres of arbitrarily large Morse index

Showed Morse index cannot be bounded by volume and topology in 4D

Contrasted with 3D compactness results by Choi-Schoen

Abstract

We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bound in terms of the volume and the topological invariants of the hypersurface itself by presenting a method for constructing Riemannian metrics on S^4 that admit embedded minimal hyperspheres of uniformly bounded volume and arbitrarily large Morse index. The phenomena we exhibit are in striking contrast with the three-dimensional compactness results by Choi-Schoen.