I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says 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 all two-sided and have multiplicity one.

In particular, I will discuss three main parts of the proof, including: the formulation of multiple-parameter min-max construction for hypersurfaces with prescribed mean curvature (PMC), an approximation scheme (by PMC min-max theory) of min-max construction of minimal hypersurfaces for relative homotopy class of boundaries, and a topological argument to reduce min-max construction for free homotopy class of mod-2 cycles to that for relative homotopy class of boundaries.

# Multiplicity One Conjecture in Min-max theory

Xin Zhou

University of California, Santa Barbara; Member, School of Mathematics

March 19, 2019