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Index estimates for free boundary minimal hypersurfaces
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Ambrozio, Lucas, Carlotto, Alessandro and Sharp, Benjamin G. (2018) Index estimates for free boundary minimal hypersurfaces. Mathematische Annalen, 370 (3-4). pp. 1063-1078. doi:10.1007/s00208-017-1549-8
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Official URL: https://doi.org/10.1007/s00208-017-1549-8
Abstract
We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser-Schoen and Folha-Pacard Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Hypersurfaces, Riemannian manifolds | ||||||||
| Journal or Publication Title: | Mathematische Annalen | ||||||||
| Publisher: | Springer Verlag | ||||||||
| ISSN: | 0025-5831 | ||||||||
| Official Date: | April 2018 | ||||||||
| Dates: |
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| Volume: | 370 | ||||||||
| Number: | 3-4 | ||||||||
| Page Range: | pp. 1063-1078 | ||||||||
| DOI: | 10.1007/s00208-017-1549-8 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Funder: | European Research Council (ERC), Scuola normale superiore (Italy) | ||||||||
| Grant number: | Start Grant PSC and LMCF 278940 |
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