Solutions of the Ginzburg–Landau equations concentrating on codimension-2 minimal submanifolds

Marco Badran; Manuel del Pino

doi:10.1112/jlms.12851

Solutions of the Ginzburg–Landau equations concentrating on codimension-2 minimal submanifolds

Marco Badran, Manuel del Pino

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Abstract

We consider the magnetic Ginzburg–Landau equations on a closed manifold (Formula presented.) (Formula presented.) formally corresponding to the Euler–Lagrange equations for the energy functional (Formula presented.) where (Formula presented.) and (Formula presented.) is a 1-form on (Formula presented.). Given a codimension-2 minimal submanifold (Formula presented.), which is also oriented and non-degenerate, we construct solutions (Formula presented.) such that (Formula presented.) has a zero set consisting of a smooth surface close to (Formula presented.). Away from (Formula presented.), we have (Formula presented.) as (Formula presented.), for all sufficiently small (Formula presented.) and (Formula presented.). Here, (Formula presented.) is a normal frame for (Formula presented.) in (Formula presented.). This improves a recent result by De Philippis and Pigati (2022), who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense.

Original language	English
Article number	e12851
Journal	Journal of the London Mathematical Society
Volume	109
Issue number	1
Early online date	13 Jan 2024
DOIs	https://doi.org/10.1112/jlms.12851
Publication status	Published - 31 Jan 2024

Funding

The authors have been supported by Royal Society Research Professorship RP-R1-180114, United Kingdom. Marco Badran is grateful to Filippo Gaia, Jared Marx-Kuo, Alessandro Pigati and Bruno Premoselli for many useful comments and enlightening discussions related to some aspects of this work..

Funders	Funder number
Royal Society	RP-R1-180114

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1112/jlms.12851

MGL_v2-4
This is the peer reviewed version of the following article: Badran, M. and del Pino, M. (2024), Solutions of the Ginzburg–Landau equations concentrating on codimension-2 minimal submanifolds. J. London Math. Soc., 109: e12851. https://doi.org/10.1112/jlms.12851. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.
Accepted author manuscript, 746 KB

Cite this

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