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Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area
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Buzano, Reto and Sharp, Benjamin G. (2018) Qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area. Transactions of the American Mathematical Society, 370 (6). pp. 4373-4399. doi:10.1090/tran/7168 ISSN 0002-9947.
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Official URL: http://dx.doi.org/10.1090/tran/7168
Abstract
We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index, the total curvature along the sequence is quantised in terms of the total curvature of some limit surface, plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in Euclidean space - all of which are finite. Thus, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Hypersurfaces, Riemannian manifolds, Curvature | |||||||||||||||
| Journal or Publication Title: | Transactions of the American Mathematical Society | |||||||||||||||
| Publisher: | American Mathematical Society | |||||||||||||||
| ISSN: | 0002-9947 | |||||||||||||||
| Official Date: | June 2018 | |||||||||||||||
| Dates: |
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| Volume: | 370 | |||||||||||||||
| Number: | 6 | |||||||||||||||
| Page Range: | pp. 4373-4399 | |||||||||||||||
| DOI: | 10.1090/tran/7168 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
| Date of first compliant deposit: | 10 April 2018 | |||||||||||||||
| Date of first compliant Open Access: | 30 April 2018 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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