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Bubbling analysis and geometric convergence results for free boundary minimal surfaces
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Ambrozio, Lucas, Buzano, Reto, Carlotto, Alessandro and Sharp, Benjamin G. (2019) Bubbling analysis and geometric convergence results for free boundary minimal surfaces. Journal de l'École polytechnique Mathématiques, 6 . pp. 621-664. doi:10.5802/jep.102 ISSN 2429-7100.
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Official URL: http://dx.doi.org/10.5802/jep.102
Abstract
We investigate the limit behaviour of sequences of free boundary minimal hypersurfaces with bounded index and volume, by presenting a detailed blow-up analysis near the points where curvature concentration occurs. Thereby, we derive a general quantization identity for the total curvature functional, valid in ambient dimension less than eight and applicable to possibly improper limit hypersurfaces. In dimension three, this identity can be combined with the Gauss-Bonnet theorem to provide a constraint relating the topology of the free boundary minimal surfaces in a converging sequence, of their limit, and of the bubbles or half-bubbles that occur as blow-up models. We present various geometric applications of these tools, including a description of the behaviour of index one free boundary minimal surfaces inside a 3-manifold of non-negative scalar curvature and strictly mean convex boundary. In particular, in the case of compact, simply connected, strictly mean convex domains in R3 unconditional convergence occurs for all topological types except the disk and the annulus, and in those cases the possible degenerations are classified.
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 , Gauss-Bonnet theorem , Three-manifolds (Topology) | ||||||
Journal or Publication Title: | Journal de l'École polytechnique Mathématiques | ||||||
Publisher: | Ecole Polytechnique | ||||||
ISSN: | 2429-7100 | ||||||
Official Date: | 2019 | ||||||
Dates: |
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Volume: | 6 | ||||||
Page Range: | pp. 621-664 | ||||||
DOI: | 10.5802/jep.102 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 28 August 2019 | ||||||
Date of first compliant Open Access: | 26 September 2019 | ||||||
RIOXX Funder/Project Grant: |
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