Communications in Analysis and Geometry

Volume 27 (2019)

Number 5

Minimal hyperspheres of arbitrarily large Morse index

Pages: 991 – 1023



Alessandro Carlotto (Department of Pure Mathematics, Imperial College London, United Kingdom)


We show that the Morse index of a closed minimal hypersurface in a four-dimensional Riemannian manifold cannot be bounded 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.

Received 10 March 2017

Accepted 8 May 2017

Published 12 November 2019