$p$-Harmonic Maps to $S^1$ and Stationary Varifolds of Codimension 2

We study the asymptotics as $p\uparrow 2$ of stationary $p$-harmonic maps $u_p\in W^{1,p}(M,S^1)$ from a compact manifold $M^n$ to $S^1$, satisfying the natural energy growth condition $$\int_M|du_p|^p=O(\frac{1}{2-p}).$$ Along a subsequence $p_j\to 2$, we show that the singular sets $Sing(u_{p_j})$ converge to the support of a stationary, rectifiable $(n-2)$-varifold $V$ of density $\Theta_{n-2}(\|V\|,\cdot)\geq 2\pi$, given by the concentrated part of the measure $$\mu=\lim_{j\to\infty}(2-p_j)|du_{p_j}|^{p_j}dv_g.$$ When $n=2$, we show moreover that the density of $\|V\|$ takes values in $2\pi\mathbb{N}$. Finally, on every compact manifold of dimension $n\geq 2$ we produce examples of nontrivial families $(1,2)\ni p\mapsto u_p\in W^{1,p}(M,S^1)$ of such maps via natural min-max constructions.