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.
Compactness and generic finiteness for free boundary minimal hypersurfaces (II)
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abstract
Given a compact Riemannian manifold with boundary, we prove that the limit of a sequence of embedded, almost properly embedded free boundary minimal hypersurfaces, with uniform area and Morse index upper bound, always inherits a non-trivial Jacobi field. To approach this, we prove a one-sided Harnack inequality for minimal graphs on balls with many holes.
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2026 1verdicts
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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.