Surveys in Differential Geometry

Volume 22 (2017)

The space of cycles, a Weyl law for minimal hypersurfaces and Morse index estimates

Pages: 319 – 329



Fernando C. Marques (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

André Neves (Department of Mathematics, University of Chicago, Illinois, U.S.A.; and Imperial College London, United Kingdom)


In this note, prepared for the occasion of the Journal of Differential Geometry (JDG) 50th birthday Conference, we will discuss a Weyl law conjectured by Gromov and proved by the authors with Liokumovich in [13], and work of the authors ([19], [20]) on the characterization of the Morse index of minimal hypersurfaces produced by min-max methods.

The last section is an update on dramatic developments obtained since the time of the conference. This includes the proof of Yau’s Conjecture (about the existence of infinitely many minimal surfaces) for generic metrics, by establishing density of minimal hypersurfaces, obtained by the authors with Irie [11], the proof of equidistribution of minimal hypersurfaces for generic metrics by the authors with Song [21], the proof of the authors’ Multiplicity One Conjecture and Morse Index Conjecture in dimension three by Chodosh and Mantoulidis ([3]), and the full resolution of Yau’s Conjecture by Song [26].

The first author was partly supported by NSF-DMS-1509027. The second author was partly supported by ERC-2011-StG-278940 and EPSRC Programme Grant EP/K00865X/1.

Published 13 September 2018