Existence of Min-Max Free Boundary Disks Releasing the Width of a Manifold

TL;DR

This paper establishes the existence of free boundary minimal disks that realize the min-max width of a manifold, using advanced harmonic map techniques and a generalized replacement procedure.

Contribution

It introduces a new replacement method and proves convexity of energy for free boundary harmonic maps, enabling the construction of minimal disks with prescribed boundary conditions.

Findings

Existence of min-max free boundary disks in manifolds.

Development of a generalized replacement procedure.

Proof of convexity of free boundary harmonic map energy.

Abstract

We perform a replacement procedure in order to produce a free boundary minimal surface whose area achieves the min-max value over all disk sweepouts of a manifold whose boundary lie in a submanifold. Our result is based on a proof of the convexity of the energy for free boundary harmonic maps and a generalization of Colding-Minicozzi replacement procedure.