Existence of at least one embedded free boundary minimal disk is shown in any mean-convex 3-ball, with at least three in strictly convex nonnegative-Ricci cases, via a multiplicity-one theorem for free boundary Simon-Smith min-max theory.
Schulz,Topological control for min-max free boundary minimal surfaces(2023), available at arXiv:math/2307.00941
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Existence of free boundary minimal disks in convex regions
Existence of at least one embedded free boundary minimal disk is shown in any mean-convex 3-ball, with at least three in strictly convex nonnegative-Ricci cases, via a multiplicity-one theorem for free boundary Simon-Smith min-max theory.