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

TL;DR

This paper proves that for any admissible family of surface immersions in a closed Riemannian manifold, the min-max area is achieved by a smooth minimal surface with multiplicity one and bounded Morse index.

Contribution

It establishes the multiplicity one property for min-max minimal surfaces in arbitrary codimension, extending previous results to a more general setting.

Findings

Min-max width is achieved by a smooth minimal surface with multiplicity one.

The Morse index of the minimal surface is bounded by the family dimension.

The result applies to arbitrary codimension immersions.

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 $k$.