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

“…We claim that for every $$\Psi \in L^2(\mathbb {R}^n)$$ we can find a solution to the system $$$$\begin{equation} {\begin{cases} -\Delta _{t,U_0}\Phi -\varepsilon ^2\Delta _{y}\Phi +\Phi +T_{U_0}\Phi =\Psi +b^\alpha (y)\mathsf {V}_\alpha (t)&\text{on }\mathbb {R}^n\\[6pt] \int _{\mathbb {R}^2}\Phi (y,t)\cdot \mathsf {V}_\alpha (t)dt=0&\alpha =1,2. \end{cases}} \end{equation}$$$$This result has already been established in [2, Lemma 7]. We sketch the proof for completeness.…”

“…Define the global approximation $$W$$ to a solution of () as $$$$\begin{equation*} W=\zeta _3W_1+(1-\zeta _3)\bm{\Psi }. \end{equation*}$$$$A lengthy but straightforward calculation similar to that of [2, section 5.5] shows that $$$$\begin{equation*} S(W)=\zeta _3S(W_1)+(1-\zeta _3)S(\bm{\Psi })+\mathrm{E}, \end{equation*}$$$$where $$$$\begin{equation*} |\mathrm{E}|\leqslant C\chi _{\lbrace 0<\zeta _3<1\rbrace }(1+|W_1|^2)(|W_1-\bm{\Psi }|+|D(W_1-\bm{\Psi })|+|\nabla ^{A_1}u_1|). \end{equation*}$$$$Now, by construction of $$\Psi$$ and the properties of the function $$f$$ and $$a$$…”

“…Gauge invariance creates an infinite dimensional part of the kernel of the linearised operator $$S^{\prime }(W)$$ around any solution $$W$$ to $$S(W)=0$$, generated by $$$$\begin{equation} \Theta _W[\gamma]=\def\eqcellsep{&}\begin{pmatrix} iu\gamma \\[2pt] {d\gamma } \end{pmatrix} \end{equation}$$$$and where again $$\gamma$$ varies among all smooth functions defined on $$N$$, see [2]. By computing the formal adjoint of $$\Theta _W$$ we can characterise orthogonality to all elements in the form (): if $$\Phi =(\phi, \omega)$$…”

“…and where again 𝛾 varies among all smooth functions defined on 𝑁, see [2]. By computing the formal adjoint of Θ 𝑊 we can characterise orthogonality to all elements in the form (1.10…”

“…2  𝛽 [ℎ]𝖵 𝛽 (𝑡) + 𝜀3  𝛽 𝑖𝑗  𝛾 𝑗𝑘   (ℎ) = 𝑓 on 𝑀 admits a solution ℎ = (𝑓) satisfying ‖ℎ‖ 𝐶 2,𝛾 (𝑀) ⩽ 𝐶‖𝑓‖ 𝐶 0,𝛾 (𝑀) .Lemma 2.1 guarantees the existence of a bounded ℎ 0 (𝑦) with  (ℎ 0 ) = −(𝗊 1 , 𝗊 2 ) 𝑇 , on 𝑀. (2.21)Choosing ℎ = 𝜀ℎ 0 , the right-hand side of (2.19) vanishes for 𝛼 = 1, 2.…”