A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities

Gareth P. Alexander, Thomas J Machon

Research output: Contribution to journalArticle (Academic Journal)peer-review

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.
Original languageEnglish
Article number20190903
JournalProceedings of the Royal Society A: Mathematical and Physical Sciences
Volume476
DOIs
Publication statusPublished - 24 Jun 2020

Fingerprint Dive into the research topics of 'A Björling representation for Jacobi fields on minimal surfaces and soap film instabilities'. Together they form a unique fingerprint.

Cite this