论文标题
四维金茨堡 - 兰道方程和编码2的整个解决方案2
Entire solutions to 4-dimensional Ginzburg-Landau equations and codimension 2 minimal submanifolds
论文作者
论文摘要
我们考虑$ \ mathbb {r}^4 $ $ $ $ $ \ begin {cases} - \ varepsilon^2（\ nabla-ia）^2U = \ frac {1} {1} {2} {2} {1- | U | U | U |^{2}） \ langle（\ nabla-ia）u，iu \ rangle \ engle \ end {cases} $$正式对应于能量功能$$的Euler-Lagrange方程 e（u，a）= \ frac {1} {2} \ int _ {\ mathbb {r}^4} |（\ nabla-ia）u |^{2}+\ va repsilon^2 | da |^{2}+\ frac {1} {4 \ varepsilon^2}（1- | u |^{2}）^{2}。 $$这里$ u：\ mathbb {r}^4 \ to \ mathbb {c} $，$ a：\ mathbb {r}^4 \ to \ mathbb {r}^4 $ and $ d $表示外部衍生作用，以$ a $ a $ a $ a $ a $ a $。给定二维最小表面$ m $ in $ \ mathbb {r}^3 $具有有限的总曲率和非脱位，我们构造了一个解决方案$（u_ \ varepsilon，a_ \ varepsilon）$，其零由零组成，由光滑的2维表面接近$ m \ m \ m \ m \ m \ m \ m \ m \ c} cy \ Mathbb {r}^4 $。远离后一个表面，我们有$ | u_ \ varepsilon | \to 1$ and $$ u_\varepsilon(x)\, \to\, \frac {z}{|z|},\quad A_\varepsilon(x)\, \to\, \frac 1{|z|^2} ( -z_2 ν(y) + z_1 {\textbf{e}}_4), \ Quad x = y +z_1ν（y） + z_2 {\ textbf {e}} _ 4 $$对于所有足够小的$ z \ ne 0 $。这里的$ y \ in m $，$ν（y）$是单位普通向量字段至$ m $ in $ \ mathbb {r}^3 $。
We consider the magnetic Ginzburg-Landau equations in $\mathbb{R}^4$ $$ \begin{cases} -\varepsilon^2(\nabla-iA)^2u = \frac{1}{2}(1-|u|^{2})u,\\ \varepsilon^2 d^*dA = \langle(\nabla-iA)u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{\mathbb{R}^4}|(\nabla-iA)u|^{2}+\varepsilon^2|dA|^{2}+\frac{1}{4\varepsilon^2}(1-|u|^{2})^{2}. $$ Here $u:\mathbb{R}^4\to \mathbb{C}$, $A: \mathbb{R}^4\to\mathbb{R}^4$ and $d$ denotes the exterior derivative acting on the one-form dual to $A$. Given a 2-dimensional minimal surface $M$ in $\mathbb{R}^3$ with finite total curvature and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ which has a zero set consisting of a smooth 2-dimensional surface close to $M\times \{0\}\subset \mathbb{R}^4$. Away from the latter surface we have $|u_\varepsilon| \to 1$ and $$ u_\varepsilon(x)\, \to\, \frac {z}{|z|},\quad A_\varepsilon(x)\, \to\, \frac 1{|z|^2} ( -z_2 ν(y) + z_1 {\textbf{e}}_4), \quad x = y + z_1 ν(y) + z_2 {\textbf{e}}_4 $$ for all sufficiently small $z\ne 0$. Here $y\in M$ and $ν(y)$ is a unit normal vector field to $M$ in $\mathbb{R}^3$.
