On the Multiplicity One Conjecture in min-max theory

• 成果类型: Article
• 署名作者: Zhou, Xin
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.192.3.3
• 发表日期: 2020
• 页码: 767-820
• 关键词: constant mean-curvature; minimal hypersurfaces; MAXIMUM PRINCIPLE; EXISTENCE; compactness; VARIETIES

摘要

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.