Abstract
We consider the magnetic Ginzburg–Landau equations in a compact manifold N
Here \(u:N\rightarrow \mathbb {C}\) and A is a 1form on N. We discuss some recent results on the construction of solutions exhibiting concentration phenomena near prescribed minimal, codimension 2 submanifolds corresponding to the vortex set of the solution. Given a codimension2 minimal submanifold \(M\subset N\) which is also oriented and nondegenerate, we construct a solution \((u_{\varepsilon },A_{\varepsilon })\) such that \(u_\varepsilon \) has a zero set consisting of a smooth surface close to M. Away from M we have
as \(\varepsilon \rightarrow 0\), for all sufficiently small \(z\ne 0\) and \(y\in M\). Here, \(\{\nu _1,\nu _2\}\) is a normal frame for M in N. These results improve, by giving precise quantitative information, a recent construction by De Philippis and Pigati (arXiv:2205.12389, 2022) who built solutions for which the concentration phenomenon holds in an energy, measuretheoretical sense. In addition, we consider the noncompact case \(N=\mathbb {R}^4\) and the special case of a twodimensional minimal surface in \(\mathbb {R}^3\), regarded as a codimension 2 minimal submanifold in \(\mathbb {R}^4\), with finite total curvature and nondegenerate. We construct a solution \((u_\varepsilon ,A_\varepsilon )\) which has a zero set consisting of a smooth 2dimensional surface close to \(M\times \{0\}\subset \mathbb {R}^4\). Away from the latter surface we have \(u_\varepsilon  \rightarrow 1\) and asymptotic behavior as in (1).
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1 Introduction
Let (N, g) be a closed Riemannian manifold of dimension \(n\ge 3\). For \(\varepsilon >0\), the magnetic Ginzburg–Landau energy on N is given by
where A is the magnetic potential, represented as a 1form \(A\in {\varOmega }^1(N)\). In (1.1) we denoted with d the exterior derivative and \(\nabla ^A:= diA\). Explicitly, the quantities involved are
where g is the metric of the manifold N. The corresponding equations are given by
where \(d^*\) is the \(L^2\)adjoint of d and \(\langle z,w\rangle =\textrm{Re}(z\bar{w})\). Explicitly, the operators in (1.2) read
where we used Einstein’s summation convention on repeated indices. Energy (1.1) models phenomena of superconductivity in presence of a magnetic field, where the regions in which \(u\approx 1\) represent portions of the material in superconducting state, while where \(u\approx 0\) the material is in its normal state. This phase transition shares many similarities to that famously described by the Allen–Cahn equation
In particular, both models exhibit concentration for solutions as the scaling parameter \(\varepsilon \rightarrow 0\). This means that the energy densities of solutions concentrate their mass (as measures) around a minimal submanifold (more generally, a rectifiable stationary varifold). Such limiting object has codimension 1 in the Allen–Cahn case and codimension 2 in the Ginzburg–Landau case.
A natural question to ask is wether or not the converse holds true.
Question 1 Given a minimal submanifold M of some ambient space N can we construct a family of solutions concentrating around M?
If N is a compact manifold and \(M\subset N\) a separating hypersurface, Pacard and Ritoré [26] proved that the answer is positive for the Allen–Cahn equation (1.3) under the assumption of nondegeneracy of M. In the case \(M=\mathbb {R}^3\), a similar result was found in [11] associated to a nondegenerate minimal surface without boundary, complete with final total curvature. In this paper, we review very recent results parallel to those in [11, 26] in the codimension 2 case for the Ginzburg–Landau equation (1.2). We present a detailed summary of the proofs whose full versions can be found in our preprints [1, 2].
Energy (1.1) is invariant under the action of the unitary group U(1), namely
In the case \(N=\mathbb {R}^2\) with \(\varepsilon =1\) it is well known that there is a unique (up to gauge transformations) degree 1 radial solution \(U_0=(u_0,A_0)^T\), with
for which \(f(0)=a(0)=0\). As established in [15, 35], the solution \(U_0\) is linearly stable. The asymptotic profile as \(r\rightarrow \infty \) is given by
We find solutions \(U_\varepsilon =(u_\varepsilon ,A_\varepsilon )\) of (1.2) concentrating as \(\varepsilon \rightarrow 0\) around a 2codimensional minimal submanifold \(M\subset N\). These solutions look like \(\varepsilon \)scalings of \(U_0\) in a region close to M in the following sense: let \(\{\nu _1,\nu _2\}\) be an orthonormal basis for \(T^\perp M\). We describe a neighbourhood of M in N by Fermi coordinates
for some \(\tau >0\). We construct solutions \(U_\varepsilon (x)=(u_\varepsilon (x),A_\varepsilon (x))\) with asymptotic behaviour given by
We do this in two different settings.
1.1 The Compact Case
We consider first a closed, ndimensional manifold N and a closed \((n2)\)dimensional minimal submanifold \(M\subset N\). We say that a minimal manifold \(M\subset N\) is admissible if

(H)
M is the boundary of a \((n1)\)dimensional, oriented, embedded submanifold \(B^{n1}\subset N^n\).
Recall that the Jacobi operator of M is the second variation of the area functional, which explicitly is given by \(\mathcal {J}[h]=(\mathcal {J}^1[h],\mathcal {J}^2[h])\), where
In the above expression R is the curvature tensor and A is the second fundamental form. We require that M is nondegenerate in the sense that the Jacobi operator has trivial bounded kernel, namely
Assumption (1.7) of nondegeneracy and Fredholm alternative for elliptic operators imply the following result.
Lemma 1.1
Let \(f\in C^{0,\gamma }(M,\mathbb {R}^2)\). Then the system
admits a solution \(h=\mathcal {H}(f)\) satisfying
Our first result is the following.
Theorem 1
([2]) Let (N, g) be a closed ndimensional Riemannian manifold and let \(M\subset N\) be an admissible, nondegenrate, codimension2 minimal submanifold. Then there is \(\delta >0\) such that for \(\sigma \in (0,1)\) and all sufficiently small \(\varepsilon >0\) there exists a solution \((u_\varepsilon ,A_\varepsilon )\) to (1.2) which as \(\varepsilon \rightarrow 0\) satisfies
for all points \(x=X(y,z)\) of the form (1.6) and where \(h_0\) is a smooth function on M. Moreover, \(u_\varepsilon \rightarrow 1\) uniformly on compact subsets of \(N\setminus M\).
1.2 The Noncompact Case
Consider the class of complete, minimal surfaces embedded in \(\mathbb {R}^3\) and with finite total curvature, that is
where K is the Gaussian curvature of M. It is known, see [18, 25, 33], that outside a large cylinder a general manifold M in this class decomposes into the disjoint union of m unbounded connected components \(M_1,\dots ,M_m\), called its ends, which are asymptotic to either catenoids or planes with parallel axes. After a rotation, we can choose coordinates \(x=(x_1,x_2,x_3)=(x',x_3)\) in \(\mathbb {R}^3\) and a large number \(R_0\) such that
Each end \(M_k\) can be represented by
where
for some constants \(a_k\), \(b_k\), \(b_{ik}\), such that the coefficients \(a_k\) are ordered and balanced, in the sense that
Given M in this class we find solutions to (1.2) when \(N=\mathbb R^4\) concentrating around \(M\times \{0\}\). As for the compact case a form of nondegeneracy is needed. However, the symmetries of the immersion of \(M\times \{0\}\) in \(\mathbb {R}^4\) automatically generate bounded Jacobi fields, given by
where
being \(\nu \) a choice of unit normal vector field on M and \(y\in M\). Thus, the nondegeneracy condition in the noncompact case becomes
We need a further geometrical condition before stating the theorem. Let \((\lambda _1,\dots ,\lambda _m)\) be a balanced, ordered vector of real numbers
and assume that, for some \(\sigma \in (0,1)\) to be determined
We are now ready to state the main result in the noncompact case.
Theorem 2
([1]) Let M be a complete, minimal surface embedded in \(\mathbb {R}^3\) and with finite total curvature and whose ends are represented by (1.9). Then there is a number \(\delta >0\) such that for all sufficiently small \(\varepsilon >0\), \(\sigma \in (0,1)\) and \(\lambda =(\lambda _1,\dots ,\lambda _m)\) satisfying conditions (1.11) and (1.12) there exists a solution \((u_\varepsilon ,A_\varepsilon )\) to (1.2) with \(N=\mathbb {R}^4\) which as \(\varepsilon \rightarrow 0\) satisfies
for all points
Here, the smooth function \(h_0: M\rightarrow \mathbb {R}^2\) satisfies
Besides, \(u_\varepsilon \rightarrow 1\) uniformly in compact subsets of \(\mathbb {R}^4\setminus M\).
Remark 1
Consider the family \(u_\varepsilon (x)\) predicted by Theorem 1. Using (1.8) we see that locally around M the equation \(u_\varepsilon =0\) has the form
where \(\theta \) is a regular function. By the implicit function theorem and the fact that \(\partial _zu_\varepsilon (0)\ne 0\) we find that locally around M the zero level set of \(u_\varepsilon \) can be parametrized by
Also, using that \(u_\varepsilon \) doesn’t vanish away from M we find that the parametrization (1.14) defines the entire 0 level set. The same argument can also be applied in the setting of Theorem 2.
There is a rich literature devoted to the connection between critical points of Allen–Cahn (resp. Ginzburg–Landau) energy and minimal submanifolds of codimension 1 (resp. 2). Concentration phenomena on minimal hypersurfaces for local minimizers of the Allen–Cahn energy have been studied in [19, 23, 24, 34], then generalized to the case of general critical points in [16] with the limiting object being a stationary varifold (a measure theoretic, nonregular generalization of minimal manfiold). The connection between solutions to the inhomogeneus Allen–Cahn equation and constantmeancurvature hypersurfaces has been studied in [31]. The reverse problem, namely the construction of concentrating families of solution has been explored, among other works, in [1, 2, 10,11,12, 26]. This concentration phenomenon has also been used as a PDE alternative to the Almgren–Pitts minmax approach [22, 28, 32] for the construction of minimal (resp. CMC) submanifolds of codimension 1, see [3, 6, 13, 14].
Similar results in the context of complexvalued Ginzburg–Landau equations have been obtained, among others, by [5, 7, 8, 17, 20, 21, 27, 30].
Recently, De Philippis and Pigati [9] established a result that complements the findings in [27] for the scenario of a nondegenerate codimension 2 minimal submanifold. Their method, based on variational techniques, does not provide detailed asymptotic information. However, they have successfully resolved the more challenging case of Ginzburg–Landau equations where no induced magnetic field is present. Our techniques do not extend to cover that particular case.
1.3 The Linearized Operator
We start by defining an inner product on the pairs \(W=(u,A)\)
where \(\langle u_1,u_2\rangle =\textrm{Re}(u_1\bar{u}_2)\) and \(A_1\cdot A_2=g^{ij}(A_1)_i(A_2)_j\), being g the metric on N. Let
If W is a solution to (1.2), i.e. \(S(W)=0\), gaugeinvariance (1.4) implies the existence of an infinite dimensional subspace of the kernel of the linearised operator \(S'(W)\) around a pair W. It’s easy to check that the gaugekernel is given by the range of \({\varTheta }_W\), where
It is also direct to check that \(L^2\)orthogonality with the space generated by \({\varTheta }_W[\gamma ]\) is characterised by it’s adjoint: if \({\varPhi }=(\phi ,\omega )\)
We recall the decomposition of the linearised operator, given by
where
is an elliptic operator which is well defined in the space of pairs \((\phi ,\omega )\) for which
where \(\nabla \omega \) is the LeviCivita connection applied to the 1form \(\omega \). We call \(L_W\) the “gaugecorrected linearised”. Define the operator \(\nabla _W\) and \({\varDelta }_W=\nabla _W^*\nabla _W\) as
By doing so, we can write
where
Recall the solution \(U_0=(u_0,A_0)\) defined in (1.5). We denote the gaugecorrected linearised operator around \(U_0\) with \(\varepsilon =1\) with
It is known that \(Z_{U_0} := \textrm{span}\{\textsf {V}_1,\textsf {V}_2\}\subset \textrm{ker}\,\textsf {L}\), where
Lastly, recall that the coercivity estimate (proved in [35])
for some \(c>0\), and Lax–Milgram theorem imply the validity of the following existence result.
Lemma 1.2
For any \({\varPsi }\in L^2(\mathbb {R}^2)\cap Z_{U_0}^\perp \) there exists a unique solution \({\varPhi }\in H^1_{U_0}(\mathbb {R}^2)\cap Z_{U_0}^\perp \) to
satisfying
for some \(C>0\).
2 Sketch of the Proof of Theorem 1
2.1 The First Approximation
In what follows we use indices \(i,j,k\dots \) and \(\alpha ,\beta ,\gamma ,\dots \) respectively for coordinates tangential and normal to M, while we use \(a,b,c,\dots \) to indicate all coordinates at once. More precisely
The admissibility hypothesis (H) allows us to choose canonically a basis \(\{\nu _1,\nu _2\}\) for \(T^\perp M\) by setting \(\nu _2\) as the normal to B in N and \(\nu _1\) as the vector field in TB restricted to M which is normal to TM (we can assume that \(\nu _1\) is directed towards B).
We describe a neighbourhood of M in N through Fermi coordinates, that is we consider the points \(x\in N\) such that
where \(B(0,\tau ) \subset \mathbb {R}^2\) and \(\tau \) is sufficiently small. Given a smooth function \(h=(h^1,h^2):M\rightarrow \mathbb {R}^2\) satisfying
consider the change of coordinates
Then, the neighbourhood of M can be described as the set \(\mathcal N=X_h(\mathcal {O}_h)\), where \(\mathcal {O}_h=\{(y,t)\ :\ y\in M,\ t+h(y)<\tau /\varepsilon \}\) and
On \(\mathcal {N}\) we define the first local approximation \(W_0\) by setting
where \(U_0=(u_0,A_0)^T\) is the degree 1 solution in \(\mathbb {R}^2\), given by (1.5).
This is a good first approximation if the error of \(W_0\)
is small. The computation of (2.3) relies on expressing the differential operators \({\varDelta }^A\) and \(d^*dA\) in coordinates (y, t). Using the fact that \(U_0\) is a solution of the corresponding system in \(\mathbb {R}^2\) (see [2]), we find
where (2.4) has been broken down into sizes in \(\varepsilon \), accounting also (2.1). The terms \(\textsf {V}_\beta \) are as in (1.17).
We look for a better approximation to a solution of (1.2) as a perturbation of \(W_0\), namely of the form \(W_1=W_0+{\varLambda }_1\). The error of approximation can be split in the following parts
where
where we recall that the 2dimensional linearized operator \(\textsf {L}\), given by (1.16), is an operator in the tvariable only. The largest term of \(S(W_0)\), namely that of order \(\varepsilon ^2\), is locally given by
Therefore, if we solve
the biggest part of the error in terms of \(\varepsilon \) is cancelled. Such \({\varLambda }_1\) exists thanks to Lemma 1.2, using that
Also, since the righthand side (2.5) is \(O(e^{t})\) for t large a standard barrier argument along with the fact that \(\textsf {L}\sim {\varDelta } + \textrm{Id}\) at infinity ensures that
for any \(0<\sigma <1\). Moreover, the error created
satisfies
To further improve the approximation the nondegeneracy assumption (1.7), and the consequent invertibility Lemma 1.1, are crucial. Indeed if we set \(W_2=W_1+{\varLambda }_2\) we can try to cancel the \(\varepsilon ^3\)terms in (2.4)
in the same manner as before. This can be done by virtue of Lemma 1.2 if the righthand side satisfies the orthogonality condition with \(Z_{U_0,t}\), that is if
Here, \(c=\int _{\mathbb R^2}\textsf {V}_\gamma (t)^2dt\) (independent of \(\gamma =1,2\)) and
Now, Lemma 1.1 guarantees the existence of a bounded \(h_0(y)\) with
Choosing \(h= \varepsilon h_0\), the righthand side of (2.6) vanishes for \(\gamma =1,2\), allowing us to find the sought \({\varLambda }_2\).
The problem with this procedure is that the term in the error created given by \(S'(W_1)\textsf {L}\) carries second derivatives in y which are not included in the invertibility theory of \(\textsf {L}\), which will create problems in further iterations of the process. Thus, an invertibility theory for the full linearised needs to be made, which is the content of Proposition 2.1 below. We also point out that the approximation \(W_1(y,t)\) will be sufficient for our purposes, and the function \({\varLambda }_2(y,t)\) will be part of the expansion of perturbation in the full solution.
The approximated solution found so far is defined only locally around M. To get a global approximation we extend \(W_1\) to the whole ambient space N. The idea is that of “gluing” \(W_1\) to a pure gauge, namely with a pair of the form
where \(\psi \) is a smooth \(S^1\)valued function defined away from M and that links well with \(W_1\) in a region close to M. We recall that, being a pure gauge, \(\varvec{\varPsi }\) satisfies automatically \(S(\varvec{\varPsi })=0\). The existence of such \(\psi \) is guaranteed by the admissibility hypothesis (H), by means of the following lemma, proved in [2].
Lemma 2.1
The admissibility hypothesis (H) guarantees the existence of a \(\delta >0\) and smooth function
where \(M_h=\{\exp _y(h^\beta (y)\nu _\beta (y)):y\in M\}\), such that for every \(x=X_h(y,t)\in \textrm{supp}\,\zeta _3\)
Let now \(\delta >0\) and \(\zeta \) be a smooth cutoff function such that \(\zeta (s)=1\) if \(s<1\) and \(\zeta (s)=0\) if \(s>2\). For \(m=1,2,\dots \) consider the cutoff functions defined by
We define the global approximation W to a solution of (1.2) as
For \(\sigma \in (0,1)\), it holds
where
and again \(S(\varvec{{\varPsi }})=0\).
2.2 Proof of Main Result
We look for a solution to (1.2) as a small perturbation of the global approximation W, namely we are looking for a \(\varPhi \) such that (Figs. 1, 2 and 3)
Roughly, the strategy to find such \(\varPhi \) is the following. First we write (2.9) as
where \(N(\varPhi )=S(W+\varPhi )S(W)S'(W)[\varPhi ]\). Secondly, we use a suitable invertibility theory for the gaugecorrected linearised \(L_W\) in order to solve
as a fixed point problem. Finally, we use the contraction mapping principle and the small Lipschitz character of N to find a \(\varPhi \) satisfying (2.9). By doing so, we find a solution to
We claim that in this case \(\gamma =0\). Denoting \(W=(u,A)\) and \(\varPhi =(\varphi ,\omega )\), we have by gauge invariance (1.4),
and since \(\varepsilon ^2{\varDelta }+u^2+\langle u,\varphi \rangle \) is a positive operator given that \(\varphi \) is small with respect to u, it must be \(\gamma =0\) and the claim is proved. It is thus sufficient to find \(\varPhi \) solving (2.10). However, (2.10) is not solvable, with the appropriate estimates, in absolute terms. Rather than that, we consider the corrected problem
The adjustment on the righthand side provides unique solvability in terms of \(\varPhi \) for a precise choice of \(b=(b^1,b^2)\), in the sense of the following result.
Proposition 2.1
Let \(0<\gamma <1\) and let \({\varLambda }\in C^{0,\gamma }(N)\). Then, there exists \(b\in C^{0,\gamma }(M)\) and a unique solution \(\varPhi =\mathcal {G}({\varLambda })\) to
satisfying
for some \(C>0\).
Proposition 2.1 allows us to write (2.11) as a fixed point problem
on the space
which admits a solution by the Lipschitz estimate
if A is chosen sufficiently large. The final step is to choose h suitably to make the projection \(\zeta _2b^\alpha \textsf {V}_\alpha \) in (2.11) vanish. We find an expression for \(b^\gamma \) by multiplying (2.11) by \(\zeta _4\textsf {V}_\gamma (t)\), \(\gamma =1,2\) and integrating on \(\mathbb {R}^2\).
As previously observed (cfr. (2.6)) the expansion of \(b^\gamma \) yields at first order a \(O(\varepsilon ^2)\) multiple of the \(\gamma \)th component of the Jacobi operator. Precisely, letting
the system \(b^\gamma =0\), \(\gamma =1,2\), can be written as
where \(G(h)=q_4^{1}(G_1(h),G_2(h))\) and
We can use Lemma 1.1 to restate (2.12) as a fixed point problem
To conclude the proof we use the following lemma (we refer to our work [2] for a proof).
Lemma 2.2
The map G satisfies
and
for some \(C>0\).
Thanks to Lemma 2.2 we see that by contraction mapping principle equation (2.13) admits a solution in the space
for any A sufficiently big. Also, the solution found satisfies \(h=\varepsilon h_0+O(\varepsilon ^2)\), where \(h_0\) is the unique solution to (2.7). This concludes the proof.
2.3 Proof of Proposition 2.1
To prove the invertibility theory for \(L_W\) we use the fact that, on a region close to M, the gaugecorrected linearised operator can be approximated by \(L_{U_0}\), namely the scaled gaugecorrected linearised operator on \(M\times \mathbb R^2\) around the canonical profile \(U_0(y,t) := U_0(t)\), which in the scaled coordinates \((y,t)=(y,z/\varepsilon )\) reads
where \(T_{U_0}\) is as in (1.15). We start by considering the cutoff functions introduced in (2.8) and looking for a solution to
of the form
where \({\varPhi }\) is defined on \(M\times \mathbb {R}^2\) and \({\varPsi }\) is defined on N. In terms of the pair \(({\varPhi },{\varPsi })\) the equation
can be broken down in the system
where we denoted
Equation (2.17) is solvable directly using the positivity of the operator on the lefthand side on \(H^1_W(N)\). For \({\varPhi }\) fixed, we find a solution \({\varPsi }\) to (2.17) satisfying
Plugging such \({\varPsi }={\varPsi }({\varLambda },{\varPhi })\) inside of (2.16) we reduce the problem to an equation for \({\varPhi }\) only. Define
Here \(\tilde{\textsf {B}}\) satisfies
where \(\delta \) is the one from the definition of \(\zeta _4\) in (2.8). With this notation equation (2.16) is equivalent to
Define the weighted norm \(C^{0,\gamma }_\sigma \) on functions \(\psi (y,t)\) defined on \(M\times \mathbb R^2\) as
where \(k\ge 0\) and \(\gamma ,\sigma \in (0,1)\). The following result holds.
Proposition 2.2
Let \(\gamma \in (0,1)\) and \(\sigma >0\) sufficiently small. Then for every \(\tilde{\varLambda }\in C^{0,\gamma }_\sigma (M\times \mathbb R^2)\) there exists \(b\in C^{0,\gamma }(M)\) such that the problem
admits a unique solution \({\varPhi }=\mathcal {T}(\tilde{\varLambda })\) satisfying
for some \(C>0\).
To prove this result we restrict to an open cover \(\{\mathcal {U}_k\}\) of M and solve the problem locally on \(\mathcal {U}_k\times \mathbb {R}^2\) for every k, finding then a global solution by gluing of all the local solutions. See [2] for details. Using Proposition 2.2 we can rephrase (2.19) as
where
Now, using that
we find
and hence by picking \(\varepsilon \), \(\delta \) sufficiently small we find a unique solution to (2.20), from which we get the existence of a unique solution \(({\varPhi },{\varPsi })\) to system (2.16)–(2.17). In conclusion, \(\varPhi =\zeta _2{\varPhi }+{\varPsi }\) solves (2.15) and it follows directly that
The proof of Proposition 2.1 is complete. \(\square \)
3 Sketch of the Proof of Theorem 2
As already mentioned, the proof of Theorem 2 follows the same lines of the proof of Theorem 1. The first main difference is that the solution’s 0 level set will actually depart logarithmically from the manifold around the ends, according to (1.13).
This fact prevents the formation of small but nondecaying errors in the space between two consecutive ends, see [11]. This fact is formulated in the following way: we choose \(h=h_*+h_1\), with
where the norms \(\Vert \cdot \Vert _{C^{0,\gamma }_\mu (M)}\) are defined by
and account for the decay along M. The function \(h_*=(h_*^1,h_*^2)^T\) satisfies \(\mathcal {J}(h_*)=0\) and
where \(M_j\) is the jth end of M. The local approximation of a solution is then given by
The fact that \(\mathcal {J}(h_*)=0\) will allow us to formulate the final, reduced problem as a fixed point involving only the Jacobi operator of \(h_1\), namely of the form
similar to (2.12). In this geometrical setting, the Jacobi operator is given by
and we recall that we assumed nondegeneracy, which here means
where \(\varvec{z}_k\), \(k=0,\dots ,4\) are given by (1.10). The presence of this kernel creates an obstruction to a direct invertibility theory for \(\mathcal {J}\). Instead, we consider the corrected problem
for some constants \(c^0,\dots ,c^4\). The correction on the righthand side provides unique solvability for (3.1) in the sense of the following lemma.
Lemma 3.1
Let \(f=(f^1,f^2)^T\) be a function defined on M such that \(\Vert f\Vert _{C^{0,\gamma }_{4}(M)}<+\infty \). Then, there exist constants \(c^0,\dots ,c^4\) such that the system
admits a solution \(h=(h^1,h^2)^T=\mathcal {H}(f)\) satisfying
With the aid of Lemma 3.1 we formulate (3.1) as a fixed point problem
which admits a solution by contraction mapping principle, using also the Lipschitz character of G (coming from calculations similar to that of the compact case, see [1]). To conclude the proof of Theorem 2 we only need to show that \(c^k=0\) for every k. This is a consequence of the fact that the Jacobi fields \(\varvec{z}_k\) are generated by the symmetries of the ambient manifold. In what follows we consider coefficients \(d_{il}\) and linear combinations
such that
At this we have constructed a solution \(U=W+\varPhi \) of
where we set \(q=\varepsilon ^2\zeta _4\tilde{q}\) and \(0<c\le \tilde{q}\le C\). Consider the quantities
It holds
This follows from gaugeinvariance, the invariances of M under rigid motions and from the balancing condition (1.11). Using (3.2) it is possible to show that the vector \((c^0,c^1,c^2,c^3,c^4,\gamma )\) is mapped to zero by a linear operator which, for \(\varepsilon \) small enough, is positive (see [1]). This implies that all coefficients vanish, and thus that we found a true solution. The proof is concluded.
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The authors have been supported by Royal Society Research Professorship RPR1180114, United Kingdom.
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Badran, M., del Pino, M. Solutions to the Magnetic Ginzburg–Landau Equations Concentrating on Codimension2 Minimal Submanifolds. Vietnam J. Math. 52, 967–984 (2024). https://doi.org/10.1007/s10013024006804
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DOI: https://doi.org/10.1007/s10013024006804