Entire solutions of the magnetic Ginzburg-Landau equation in $\mathbb{R}^4$

Liu, Yong; Ma, Xiang; Wei, Juncheng; Wu, Wangze

doi:10.48550/arxiv.2108.02754

Cited by 3 publications

(3 citation statements)

References 21 publications

(33 reference statements)

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“…Our result can be interpreted as a form of converse of their statement in the (non-compact) entire space for a special class of minimal submanifolds. In the recent work [23] Liu, Ma, Wei and Wu have found a different class of entire solutions to (1.5) connected with the so-called saddle solution of Allen-Cahn equation and a special minimal surface built by Arezzo and Pacard [1]. Symmetries allow them to reduce the problem to one in two variables.…”

Section: Resultsmentioning

confidence: 99%

Entire solutions to 4-dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds

Badran¹,

Pino²

2022

Preprint

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We consider the magnetic Ginzburg-Landau equations informally corresponding to the Euler-Lagrange equations for the energy functionalHere u : R 4 → C, A : R 4 → R 4 and d denotes exterior derivative when A is regarded as a 1-form in R 4 . Given a 2-dimensional minimal surface M in R 3 with finite total curvature and non-degenerate, we construct a solution (uε, Aε) which has a zero set consisting of a smooth 2-dimensional surface close to M × {0} ⊂ R 4 . Away from the latter surface we have |uε| → 1 andfor all sufficiently small z = 0. Here y ∈ M and ν(y) is a unit normal vector field to M in R 3 .

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Section: Resultsmentioning

confidence: 99%

Entire solutions to 4-dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds

Badran¹,

Pino²

2022

Preprint

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“…We have recently used this approach in a similar context in ℝ 4 obtaining concentration for a special class of non-compact minimal surfaces embedded in ℝ 3 that includes a catenoid and the Costa-Hoffman-Meeks minimal surfaces [2]. Liu, Ma, Wei and Wu [22] have obtained the existence of a solution in ℝ 4 with precise asymptotics and concentration on a special non-compact codimension-2 symmetric minimal manifolds discovered by Arezzo and Pacard [1]. See also [10] for a recent construction in ℝ 3 of interacting helicoidal vortex filaments in Ginzburg-Landau without magnetic field.…”

Section: Introductionmentioning

confidence: 99%

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

Badran,

del Pino

2024

Journal of London Math Soc

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We consider the magnetic Ginzburg–Landau equations on a closed manifold formally corresponding to the Euler–Lagrange equations for the energy functional where and is a 1‐form on . Given a codimension‐2 minimal submanifold , which is also oriented and non‐degenerate, we construct solutions such that has a zero set consisting of a smooth surface close to . Away from , we have as , for all sufficiently small and . Here, is a normal frame for in . 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.

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“…We have recently used this approach in a similar context in R 4 obtaining concentration for a special class of non-compact minimal surfaces embedded in R 3 which includes a catenoid and the Costa-Hoffman-Meeks minimal surfaces [2]. Liu, Ma, Wei and Wu [19] have obtained the existence of a solution in R 4 with precise asymptotics and concentration on a special non-compact codimension-2 symmetric minimal manifolds discovered by Arezzo and Pacard [1]. See also [9] for a recent construction in R 3 of interacting helicoidal vortex filaments in Ginzburg-Landau without magnetic field.…”

Section: Introductionmentioning

confidence: 99%

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

Badran¹,

Pino²

2022

Preprint

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We consider the magnetic Ginzburg-Landau equations in a compact manifold Nformally corresponding to the Euler-Lagrange equations for the energy functionalHere u : N → C and A is a 1-form on N . Given a codimension-2 minimal submanifold M ⊂ N which is also oriented and non-degenerate, we construct a solution (uε, Aε) such that uε has a zero set consisting of a smooth surface close to M . Away from M we haveas ε → 0, for all sufficiently small z = 0 and y ∈ M . Here, {ν 1 , ν 2 } is a normal frame for M in N . This improves a recent result by De Philippis and Pigati [10] who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense.We consider a closed, n-dimensional manifold N n and a closed n − 2-dimensional minimal submanifold M n−2 ⊂ N n . We say that a minimal manifold. We also require that M is non-degenerate, in the sense that the Jacobi operator has trivial bounded kernel, namely(1.6)We recall that the Jacobi operator is the second variation of the area functional around M , namelyNow we state the main result.

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Entire solutions of the magnetic Ginzburg-Landau equation in $\mathbb{R}^4$

Cited by 3 publications

References 21 publications

Entire solutions to 4-dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds

Entire solutions to 4-dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds

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

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

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