Quantization and non-quantization of energy for higher-dimensional Ginzburg-Landau vortices

Abstract

Given a family of critical points $u_ε:M^n\to\mathbb{C}$ for the complex Ginzburg--Landau energies \begin{align*} &E_ε(u)=\int_{M}\left(\frac{|du|^2}{2}+\frac{(1-|u|^2)^2}{4ε^2}\right), \end{align*} on a manifold $M$, with natural energy growth $E_ε(u_ε)=O(|\logε| )$, it is known that the vorticity sets $\{|u_ε|\leq \frac{1}{2}\}$ converge subsequentially to the support of a stationary, rectifiable $(n-2)$-varifold $V$ in the interior, characterized as the concentrated portion of the limit $\lim_{ε\to 0} \frac{e_ε(u_ε)}{π|\logε| }$ of the normalized energy measures. When $n=2$ or the solutions $u_ε$ are energy-minimizing, it is known moreover that this varifold $V$ is integral; i.e., the $(n-2)$-density $Θ_{n-2}(|V|,x)$ of $|V|$ takes values in $\mathbb{N}$ at $|V|$-a.e. $x\in M$. In the present paper, we show that for a general family of critical points with $E_ε(u_ε)=O(|\logε| )$ in dimension $n\geq 3$, this energy quantization phenomenon only holds where the density is less than $2$: namely, we prove that the density $Θ_{n-2}(|V|,x)$ of the limit varifold takes values in $\{1\}\cup [2,\infty)$ at $|V|$-a.e. $x\in M$, and show that this is sharp, in the sense that for any $n\geq 3$ and $θ\in \{1\}\cup [2,\infty)$, there exists a family of critical points $u_ε$ for $E_ε$ in the ball $B_1^n(0)$ with concentration varifold $V$ given by an $(n-2)$-plane with density $θ$.