On the Multiplicity One Conjecture in min-max theory

By Xin Zhou. Annals of Mathematics, 2019

136 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7f659e7574db753f26160bd5ed6f040ba85075b2

Citations in the corpus

• Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, unknown): “Besides the one-sided components have even multiplicities: this follows from the Multiplicity One theorem [49, 45], and an elementary perturbation and compactness argument.”