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

Given a family of critical points $u_{\epsilon}:M^n\to\mathbb{C}$ 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 $E_{\epsilon}(u_{\epsilon})=O(\lvert\log\epsilon\rvert)$, it is known that the vorticity sets $\{|u_\epsilon|\le\frac{1}{2}\}$ converge subsequentially to the support of a stationary, rectifiable $(n-2)$-varifold $V$ in the interior, characterized as the concentrated portion of the limit $\lim_{\epsilon\to 0} \frac{e_\epsilon(u_\epsilon)}{\pi\lvert\log\epsilon\rvert}$ of the normalized energy measures.

When $n=2$ or the solutions $u_{\epsilon}$ are energy-minimizing, it is known moreover that this varifold $V$ is integral; i.e., the $(n-2)$-density $\Theta_{n-2}(|V|,x)$ 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 $E_{\epsilon}(u_{\epsilon})=O(\lvert\log\epsilon\rvert)$ in dimension $n\geq 3$, this energy quantization phenomenon only holds where the density is less than $2$: namely, we prove that the density $\Theta_{n-2}(|V|,x)$ 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 $\theta\in \{1\}\cup [2,\infty)$, there exists a family of critical points $u_{\epsilon}$ for $E_{\epsilon}$ in the ball $B_1^n(0)$ with concentration varifold $V$ given by an $(n-2)$-plane with density $\theta$.