Under positive scalar curvature and mean-convex boundary, a Morse index bound bounds area, genus, boundary components, and total curvature of free boundary minimal surfaces, while topology or area bounds alone fail to bound the others.
Bounds for the Morse index of free boundary minimal surfaces
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abstract
Inspired by work of Ejiri-Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of Fraser-Li, we conclude that the area index of a free-boundary minimal surface in a convex domain of Euclidean three-space, is bounded from above by a linear function of its genus and its number of boundary components. We also prove index bounds for submanifolds of higher dimension.
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math.DG 1years
2019 1verdicts
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Inequivalent complexity criteria for free boundary minimal surfaces
Under positive scalar curvature and mean-convex boundary, a Morse index bound bounds area, genus, boundary components, and total curvature of free boundary minimal surfaces, while topology or area bounds alone fail to bound the others.