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Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds

Marco Badran, Manuel del Pino

TL;DR

A solution is built for which the concentration phenomenon holds in an energy, measure-theoretical sense in a compact manifold.

Abstract

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.

Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds

TL;DR

A solution is built for which the concentration phenomenon holds in an energy, measure-theoretical sense in a compact manifold.

Abstract

We consider the magnetic Ginzburg-Landau equations in a compact manifold formally corresponding to the Euler-Lagrange equations for the energy functional Here and is a 1-form on . Given a codimension-2 minimal submanifold which is also oriented and non-degenerate, we construct a solution such that has a zero set consisting of a smooth surface close to . Away from we have as , for all sufficiently small and . Here, is a normal frame for in . 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.
Paper Structure (13 sections, 11 theorems, 224 equations, 3 figures)

This paper contains 13 sections, 11 theorems, 224 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The role of gauge invariance in the linearized equation
  3. The first approximation
  4. The operators in coordinates
  5. Proof of main result
  6. Proof of Proposition \ref{['inver L_W']}
  7. Invertibility of $L_{U_0}$
  8. Proof of Proposition \ref{['invert LU0']}
  9. Proofs of Lemmas \ref{['G contracts']}, \ref{['Solvability lin flat space']} and \ref{['positive operator in MR2']}
  10. Proof of Lemma \ref{['G contracts']}
  11. Proof of Lemma \ref{['Solvability lin flat space']}
  12. Proof of Lemma \ref{['positive operator in MR2']}
  13. Construction of the outer function $\psi$

Key Result

Theorem 1

Let $N$ be a closed $n$-dimensional manifold and let $M\subset N$ be an admissible, non-degenrate, codimension-2 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 equation for all points $x=X(y,z)$ of the form tub neigh and where $h_0$ is a smooth function on $M$, explic

Figures (3)

  • Figure 1: A representation of $M$ as the boundary of an oriented manifold $B$ in $N$. Assumption (H) determines the normal fields $\{\nu_1,\nu_2\}$.
  • Figure 2: The normal frame around $M$.
  • Figure 3: A cross-section of $B$ and $M=\partial B$, with a representation of the set $\mathcal{W}$ with its subset $\mathcal{T}$.

Theorems & Definitions (11)

  • Theorem 1
  • Lemma 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 3.1
  • Lemma 3.1
  • Lemma 4.1
  • Proposition 4.1
  • Lemma 5.1
  • Lemma 5.2
  • ...and 1 more