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

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.

we have just corrected some typos with respect to the previous version. This version will be published in Advances in Mathematics