Minimal hyperspheres of arbitrarily large Morse index

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.

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Received 10 March 2017

Accepted 8 May 2017

Published 12 November 2019