论文标题
Ginzburg-Landau方程的解决方案集中于Codimension-2 Minimal Submanifolds
Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds
论文作者
论文摘要
我们考虑紧凑的歧管$ n $ $ $ $ $ $ beken {case} - \ varepsilon^2Δ^{a} u = \ frac {1} {2} {2}（1- | u |^2） \ end {cases} $$正式对应于能量功能$$的欧拉 - 拉格朗日方程 e（u，a）= \ frac {1} {2} \ int_ {n} \ varepsilon^2 | \ nabla^au |^au |^{2}+\ varepsilon^4 | da | da | da |^{2} {2}+\ frac+\ frac {1} {1} {1} {4} {4} {4} {1-1- |^2}} $$这里$ u：n \ to \ mathbb {c} $，$ a $是$ n $的1形。给定一个Codimension-2最小submanifold $ m \子集n $，它也是定向和非脱位，我们构造了一个解决方案$（u_ \ varepsilon，a_ \ varepsilon）$，使得$ u_ \ u_ \ varepsilon $的平滑表面零集，由平滑的表面靠近$ m $。远离$ m $，我们有$$ u_ \ varepsilon（x）\ to \ frac {z} {| z |}，\ quad a_ \ varepsilon（x）\ to \ frac {1} {1} {| z | |^2}（ - z^2dz^2dz^1+z^1+z^1dz^2） $$ as $ \ varepsilon \ to 0 $，对于所有足够小的$ z \ ne 0 $和$ y \ in m $。在这里，$ \ {ν_1，ν_2\} $是$ n $中的$ m $的普通框架。这改善了de Philippis和Pigati的最新结果，他们建立了一种解决方案，该解决方案具有浓度现象具有衡量理论意义的能量。
We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$ $$ \begin{cases} -\varepsilon^2 Δ^{A} u=\frac{1}{2}(1-|u|^2)u,\\ \varepsilon^2 d^*dA=\langle\nabla^A u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{N}\varepsilon^2|\nabla^Au|^{2}+\varepsilon^4|dA|^{2}+\frac{1}{4}(1-|u|^{2})^{2}. $$ Here $u:N\to \mathbb{C}$ and $A$ is a 1-form on $N$. Given a codimension-2 minimal submanifold $M\subset N$ which is also oriented and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ such that $u_\varepsilon$ has a zero set consisting of a smooth surface close to $M$. Away from $M$ we have $$ u_\varepsilon(x)\to\frac{z}{|z|},\quad A_\varepsilon(x)\to\frac{1}{|z|^2}(-z^2dz^1+z^1dz^2),\quad x=\exp_y(z^βν_β(y)). $$ as $\varepsilon\to 0$, for all sufficiently small $z\ne 0$ and $y\in M$. Here, $\{ν_1,ν_2\}$ is a normal frame for $M$ in $N$. This improves a recent result by De Philippis and Pigati who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense.
