Free boundary minimal disks in convex balls

TL;DR

This paper proves the existence of multiple free-boundary minimal disks in convex 3-balls with nonnegative Ricci curvature, using a combination of geometric analysis techniques, and establishes related mean-convex foliations.

Contribution

It demonstrates the existence of at least three free-boundary minimal disks in convex 3-balls under generic metrics, extending prior work and employing novel methods.

Findings

At least 3 embedded free-boundary minimal disks exist in convex 3-balls with generic metrics.

Existence of at least 2 solutions without genericity assumptions.

Establishment of smooth free-boundary mean-convex foliations.

Abstract

In this paper, we prove that every strictly convex 3-ball with nonnegative Ricci-curvature contains at least 3 embedded free-boundary minimal 2-disks for any generic metric, and at least 2 solutions even without genericity assumption. Our approach combines ideas from mean curvature flow, min-max theory and degree theory. We also establish the existence of smooth free-boundary mean-convex foliations. In stark contrast to our prior work in the closed setting, the present result is sharp for generic metrics.