Min-max minimal disks with free boundary in Riemannian manifolds

TL;DR

This paper develops a min-max theory for constructing minimal disks with free boundary in closed Riemannian manifolds, extending classical minimal surface theory to a more general setting with new energy convexity results.

Contribution

It introduces an effective min-max framework for free boundary minimal disks, generalizing previous theories and establishing energy convexity and uniqueness for weakly harmonic maps with mixed boundary conditions.

Findings

Established energy convexity for weakly harmonic maps with free boundary.

Proved uniqueness of weakly harmonic maps with mixed boundary conditions.

Extended min-max theory to include free boundary minimal disks in Riemannian manifolds.

Abstract

In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in $\mathbf{R}^n$ to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit $2$-disk in $\mathbf{R}^2$ into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit $2$-disk proved by Colding and Minicozzi.