Comparing the Morse index and the first Betti number of minimal hypersurfaces
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.
Keywords
Cite
@article{arxiv.1601.08152,
title = {Comparing the Morse index and the first Betti number of minimal hypersurfaces},
author = {Lucas Ambrozio and Alessandro Carlotto and Ben Sharp},
journal= {arXiv preprint arXiv:1601.08152},
year = {2016}
}
Comments
accepted for publication on the Journal of Differential Geometry, minor typographical revisions and comments added