We prove that in a closed manifold of dimension between $3$ and $7$ with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves.
We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.
"On the Multiplicity One Conjecture in min-max theory." Ann. of Math. (2) 192 (3) 767 - 820, November 2020. https://doi.org/10.4007/annals.2020.192.3.3