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.
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Every closed Riemannian 4- or 5-manifold admits a branched immersed closed minimal surface, obtained from a min-max sequence of sweepouts generated by multisections and refined by harmonic replacement.
In 3-spheres with positive Ricci curvature and scalar curvature at least Lambda_0 > 0, there exist four distinct embedded minimal 2-spheres with areas at most 12 pi (i+1)/Lambda_0, plus an application showing at least three non-planar minimal spheres in suitable ellipsoids.
Closed Riemannian manifolds with compact isometric group actions contain infinitely many invariant minimal hypersurfaces, and under a finiteness assumption each G-homology class contains infinitely many distinct embedded realizations.
Existence of index-one minimal hypersurfaces with unbounded volume in enlargeable manifolds (dims 3-7) plus 3D scalar curvature rigidity under area-nonincreasing maps.
citing papers explorer
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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.
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Existence of classical minimal surfaces in $4$ and $5$-manifolds
Every closed Riemannian 4- or 5-manifold admits a branched immersed closed minimal surface, obtained from a min-max sequence of sweepouts generated by multisections and refined by harmonic replacement.
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Minimal spheres and scalar curvature
In 3-spheres with positive Ricci curvature and scalar curvature at least Lambda_0 > 0, there exist four distinct embedded minimal 2-spheres with areas at most 12 pi (i+1)/Lambda_0, plus an application showing at least three non-planar minimal spheres in suitable ellipsoids.
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Infinite existence of equivariant minimal hypersurfaces
Closed Riemannian manifolds with compact isometric group actions contain infinitely many invariant minimal hypersurfaces, and under a finiteness assumption each G-homology class contains infinitely many distinct embedded realizations.
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Closed minimal surfaces of index one in Riemannian manifolds
Existence of index-one minimal hypersurfaces with unbounded volume in enlargeable manifolds (dims 3-7) plus 3D scalar curvature rigidity under area-nonincreasing maps.