A proof of the multiplicity 1 conjecture for min-max minimal surfaces in arbitrary codimension

Abstract

Given any admissible $k$-dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity $1$ and Morse index bounded by $k$.