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

Given a family of critical points $u_{ϵ}:M^n\to \mathbb{C}$ for the complex Ginzburg–Landau energies \begin{align*} &E_\epsilon(u)=\int_{M}\left(\frac{|du|^2}{2}+\frac{(1-|u|^2)^2}{4\epsilon^2}\right), \end{align*} on a manifold $M$, with natural energy growth $E_{ϵ}(u_{ϵ})=O(\mathrm{\mid }\log ϵ\mathrm{\mid })$, it is known that the vorticity sets $\{\mathrm{\mid }{u}_{ϵ}\mathrm{\mid }\le \frac{1}{2}\}$ converge subsequentially to the support of a stationary, rectifiable $(n-2)$-varifold $V$ in the interior, characterized as the concentrated portion of the limit $\lim_{\epsilon\to 0} \frac{e_\epsilon(u_\epsilon)}{\pi|\log\epsilon| }$ 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 ${\mathrm{\Theta }}_{n-2}(\mathrm{\mid }V\mathrm{\mid },x)$ of $\mathrm{\mid }V\mathrm{\mid }$ takes values in $\mathbb{N}$ at $\mathrm{\mid }V\mathrm{\mid }$-a.e. $x\in M$. In the present paper, we show that for a general family of critical points with $E_{ϵ}(u_{ϵ})=O(\mathrm{\mid }\log ϵ\mathrm{\mid })$ in dimension $n\ge 3$, this energy quantization phenomenon only holds where the density is less than $2$: namely, we prove that the density ${\mathrm{\Theta }}_{n-2}(\mathrm{\mid }V\mathrm{\mid },x)$ of the limit varifold takes values in $\{1\}\cup [2,\mathrm{\infty })$ at $\mathrm{\mid }V\mathrm{\mid }$-a.e. $x\in M$, and show that this is sharp, in the sense that for any $n\ge 3$ and $\theta\in {1}\cup [2,\infty)$,thereexistsafamilyofcriticalpoints$u_{\epsilon}$for$E_{\epsilon}$intheball$B_1^n(0)$withconcentrationvarifold$V$ given by an $(n-2)$-plane with density $\theta$.