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.

Xin Zhou

Annals of Mathematics

《数学年鉴》由普林斯顿大学数学系与高等研究院合作出版双月刊。该杂志由弗吉尼亚大学的奥蒙德·斯通于 1884 年创办，1899 年转入哈佛大学，1911 年转入普林斯顿大学。自 1933 年以来，《年鉴》由普林斯顿大学和高等研究院联合编辑。

The Annals of Mathematics is a bi-monthly journal published in collaboration between the Department of Mathematics at Princeton University and the Institute for Advanced Study. Founded by Ormond Stone at the University of Virginia in 1884, it moved to Harvard University in 1899 and then to Princeton University in 1911. Since 1933, the journal has been jointly edited by Princeton University and the Institute for Advanced Study.

《Annals of Mathematics》是由普林斯顿大学数学系与高等研究院合作出版的双月刊。该杂志由弗吉尼亚大学的奥蒙德·斯通于1884年创办，1899年转入哈佛大学，1911年转入普林斯顿大学。自1933年以来，《Annals of Mathematics》由普林斯顿大学和高等研究院联合编辑。

Category	Quartile
数学	1
数学, 数学	1

On the Multiplicity One Conjecture in min-max theory

ivySCI AI Smartly Parses PDF, Answers Researchers' Questions, and Helps You Understand Papers in Seconds

Journal Info