A proof of the multiplicity one 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 one and Morse index bounded by $k$.

34 pages (major improvements in the presentation: several typos fixed, few proofs expanded, added explanatory Subsection 5.1)