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Free boundary minimal disks in convex balls
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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.
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Minimal surfaces with arbitrary genus in 3-spheres of positive Ricci curvature
Every positively curved Riemannian 3-sphere contains an embedded genus-g minimal surface of area at most 2 sigma_1 for every g.
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