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A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities
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Alexander, Gareth P. and Machon, Thomas (2020) A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476 (2238). 20190903. doi:10.1098/rspa.2019.0903 ISSN 1471-2946.
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Official URL: https://doi.org/10.1098/rspa.2019.0903
Abstract
We develop a general framework for the description of instabilities on soap films using the Björling representation of minimal surfaces. The construction is naturally geometric and the instability has the interpretation as being specified by its amplitude and transverse gradient along any curve lying in the minimal surface. When the amplitude vanishes, the curve forms part of the boundary to a critically stable domain, while when the gradient vanishes the Jacobi field is maximal along the curve. In the latter case, we show that the Jacobi field is maximally localized if its amplitude is taken to be the lowest eigenfunction of a one-dimensional Schrödinger operator. We present examples for the helicoid, catenoid, circular helicoids and planar Enneper minimal surfaces, and emphasize that the geometric nature of the Björling representation allows direct connection with instabilities observed in soap films.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics T Technology > T Technology (General) |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Physics | ||||||
SWORD Depositor: | Library Publications Router | ||||||
Library of Congress Subject Headings (LCSH): | Minimal surfaces, Geometry, Riemannian, Soap | ||||||
Journal or Publication Title: | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | ||||||
Publisher: | The Royal Society | ||||||
ISSN: | 1471-2946 | ||||||
Official Date: | 24 June 2020 | ||||||
Dates: |
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Volume: | 476 | ||||||
Number: | 2238 | ||||||
Article Number: | 20190903 | ||||||
DOI: | 10.1098/rspa.2019.0903 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Copyright Holders: | © 2020 The Author(s) | ||||||
Date of first compliant deposit: | 3 July 2020 | ||||||
Date of first compliant Open Access: | 6 July 2020 | ||||||
RIOXX Funder/Project Grant: |
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