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

Abstract

Given a family of critical points $u_{\epsilon }:M^n\to \mathbb{C}$ 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 $E_{\epsilon }(u_{\epsilon })=O(|\log \epsilon |)$, 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 ${\lim }_{\epsilon \to 0}\frac{e_{\epsilon }(u_{\epsilon })}{\pi |\log \epsilon |}$ 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 $E_{\epsilon }(u_{\epsilon })=O(|\log \epsilon |)$ in dimension $n\ge 3$, 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 $n\ge 3$ 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 $\theta$.

this is the version published on Ars Inveniendi Analytica