On the bifurcation theory of the GinzburgLandau equations

2025-05-02 0 0 182.96KB 15 页 10玖币

侵权投诉

arXiv:2210.03271v2 [math.AP] 7 Apr 2023

ON THE BIFURCATION THEORY OF THE GINZBURG–LANDAU EQUATIONS

ÁKOS NAGY AND GONÇALO OLIVEIRA

ABSTRACT. We construct nonminimal and irreducible solutions to the Ginzburg–Landau equa-

tions on closed manifolds of arbitrary dimension with trivial ﬁrst real cohomology. Our method

uses bifurcation theory where the “bifurcation points” are characterized by the eigenvalues of

a Laplace-type operator. To our knowledge these are the ﬁrst such examples on nontrivial line

bundles.

1. INTRODUCTION

We consider Ginzburg–Landau theory, also known as abelian Yang–Mills–Higgs theory, on

Riemannian manifolds. More concretely, if ¡X,g¢is an N-dimensional, closed, oriented, Rie-

mannian manifold, (L,h)is a Hermitian line bundle over X, and τ,κare two positive coupling

constants, then the Ginzburg–Landau (free) energy of a unitary connection ∇and smooth sec-

tion φ(both on L) is given by

(∇,φ)7→ 1

X³|F∇|2+|∇φ|2+κ2

2¡τ−|φ|2¢2´volg. (1.1)

By [12, Proposition A.1] every critical point of (1.1) is gauge equivalent to a smooth one, which

in turn is a solution to

d∗F∇−iIm(h(∇φ,φ)) =0, (1.2a)

∇∗∇φ−κ2¡τ−|φ|2¢φ=0. (1.2b)

We call equations (1.2a) and (1.2b) the Ginzburg–Landau equations. These are nonlinear, sec-

ond order, elliptic partial differential equations which are invariant by the action of the group

of automorphisms of (L,h), also known as the gauge group. If a unitary connection ∇0satis-

ﬁes the abelian Yang–Mills equation (also known as the source-free Maxwell’s equation)

d∗F∇0=0, (1.3)

then the pair ¡∇0,0¢solves the Ginzburg–Landau equations; such a pair is said to be a normal

phase solution. Notice that equation (1.3) is independent of τand κ. As is common in abelian

Date: April 10, 2023.

2020 Mathematics Subject Classiﬁcation. 35Q56, 53C07, 58E15, 58J55.

Key words and phrases. Ginzburg–Landau equations, nonminimal solutions, bifurcation theory.

gauge theories, we call a pair (∇,φ)reducible if φvanishes identically, and irreducible other-

wise. A solution to the Ginzburg–Landau equations is reducible if and only if it is a normal

phase solution.

On closed manifolds absolute minimizers of the Ginzburg–Landau energy (1.1) exists (cf;

[6,12]) which are automatically solutions Ginzburg–Landau equations (1.2a) and (1.2b). The

minimizers, often called vortices, are well-understood, especially on Kähler manifolds and

for critical coupling, that is, when κ=1

p2; cf. [1,3,5] and more recently [2,6]. The criti-

cally coupled case has special properties that the others lack, for example “self-duality” via

a Bogomolny-type trick.

Much less is known about nonminimal solutions. In [12], Pigati and Stern constructed irre-

ducible solutions on (topologically) trivial line bundles over closed Riemannian manifolds. As

these solutions have positive energy, they cannot be the absolute minima of Ginzburg–Landau

energy (1.1). Furthermore, in our companion paper [7], we constructed nonminimal critical

points on nontrivial line bundles over oriented and closed surfaces.

In this paper, motivated by and building on works of [2,6,9–11], we construct new, nonmin-

imal, and irreducible solutions to the Ginzburg–Landau equations (1.2a) and (1.2b). To the

best of our knowledge, on nontrivial bundles and in dimension greater than two, these solu-

tions are the only known nonminimal and irreducible solutions so far, and together with the

solutions of [7,12], these solutions are the only known nonminimal and irreducible solutions

on any line bundle.

Summary of main results. We construct solutions on closed manifolds satisfying certain topo-

logical/geometric conditions. The proof uses a technique inspired by Lyapunov–Schmidt re-

duction; cf. [4, Chapter 5].

Main Theorem 1. Let X be a closed, oriented, N-dimensional, Riemannian manifold with a

Hermitian line bundle L, and ¡∇0,0¢be a normal phase solution on L. Let ∆0.

.=¡∇0¢∗∇0

acting on square integrable sections of Land λ∈Spec(∆0). Assume X has trivial ﬁrst de Rham

cohomology.

Then there exists t0>0and for each t ∈(0,t0)an element Φt∈ker(∆0−λ)with unit L2-

norm such that there is a (possibly discontinuous) branch of triples

©(At,φt,τt)∈Ω1×¡L2

1∩LN¢×+|t∈(0,t0)ª, (1.4)

of the form

(At,φt,τt)=³t2At,tΦt+t3Ψt,λ

κ2+t2ǫt´,

such that the family

{ (At,Ψt,ǫt)|t∈(0,t0) },

is determined by Φtand is bounded in L2

1×¡L2

1∩LN∩¡ker(∆0−λ)⊥¢¢×+, and for each t ∈

(0,t0)the pair ¡∇0+At,φt¢is an irreducible solution to the Ginzburg–Landau equations (1.2a)

and (1.2b) with τt.

Remark 1.1. In Theorem 3.6 we consider a similar case, where we get a weaker result. Namely,

we remove the assumption that X has trivial ﬁrst de Rham cohomology, and replace it with

the conditions that κ2Ê1

2, X is Kähler, ∇0is Hermitian Yang–Mills, and Lcarries nontrivial

holomorphic sections with respect to ∇0, and λ=min¡Spec(∆0)¢. This result is a generalization

of the main result of [2], which only covered closed surfaces of high genus with line bundles of

high degree. Our result extends this to all closed Kähler manifolds and line bundles.

Organization of the paper. In Section 2, we give a brief introduction to the important geo-

metric analytic aspects of the Ginzburg–Landau theory that are needed to prove our results.

In Section 3, we use bifurcation theory to construct novel solutions to the Ginzburg–Landau

equations, in any dimensions, on closed Riemannian manifolds with zero ﬁrst Betti number.

In Section 3.1, we prove a similar result for Kähler manifolds.

Acknowledgment. The ﬁrst author is thankful to Israel Michael Sigal, Steve Rayan, and Daniel

Stern for their help with various parts of the paper.

The second author was supported by Fundação Serrapilheira 1812-27395, by CNPq grants

428959/2018-0 and 307475/2018-2, and FAPERJ through the program Jovem Cientista do Nosso

Estado E-26/202.793/2019.

The authors are thankful to the anonymous Referee for their thorough feedback.

2. GINZBURG–LANDAU THEORY ON MANIFOLDS

Let ¡X,g¢be a closed, oriented, Riemannian manifold of dimension N. Let us ﬁx a con-

nection ∇0that satisﬁes equation (1.3). We deﬁne the Sobolev norms of L-valued forms via

the Levi-Civita connection of ¡X,g¢and the connection ∇0. Note that the induced topologies

are the same for any choice of ∇0and since the moduli of normal phase solutions is compact

(modulo gauge), and if a Coulomb-type gauge ﬁxing condition is chosen (with respect to a

reference connection), then the family of norms are in fact uniformly equivalent, that is, for

all kand p, there exists a number Ck,pÊ1, such that for any two Sobolev Lp

k-norms, k·kLp

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

arXiv:2210.03271v2[math.AP]7Apr2023ONTHEBIFURCATIONTHEORYOFTHEGINZBURG–LANDAUEQUATIONSÁKOSNAGYANDGONÇALOOLIVEIRAABSTRACT.WeconstructnonminimalandirreduciblesolutionstotheGinzburg–Landauequa-tionsonclosedmanifoldsofarbitrarydimensionwithtrivialﬁrstrealcohomology.Ourmethodusesbifurcationtheorywherethe...

展开>> 收起<<

On the bifurcation theory of the GinzburgLandau equations.pdf

共15页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

On the bifurcation theory of the GinzburgLandau equations

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: