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.

中文翻译：

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极小极大理论中的多重一猜想

我们证明，在具有颠簸度量的 3 到 7 维的闭合流形中，与 Gromov、Guth、Marques-Neves 引入的体积谱相关的 min-max 最小超曲面是两侧的并且具有多重性。这证实了 Marques-Neves 的猜想。我们证明，在凹凸度量中，每个体积谱都是通过 Caccioppoli 集边界扫除的某些相对同伦类的最小值-最大值来实现的。主要结果是使用作者与朱建立的具有规定平均曲率的超曲面的最小-最大理论来近似这样的最小-最大值。