$p$-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 $${\int }_M\mathrm{\mid }d{u}_p{\mathrm{\mid }}^p=O(\frac{1}{2-p})\mathrm{.}$$ 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 ${\mathrm{\Theta }}_{n-2}(\mathrm{\parallel }V\mathrm{\parallel },\cdot )\ge 2\pi$, given by the concentrated part of the measure $$\mu =\underset{j\to \mathrm{\infty }}{\lim }(2-p_j)\mathrm{\mid }d{u}_{p_j}{\mathrm{\mid }}^{p_j}d{v}_g\mathrm{.}$$ When $n=2$, we show moreover that the density of $\mathrm{\parallel }V\mathrm{\parallel }$ takes values in $2\pi \mathbb{N}$. Finally, on every compact manifold of dimension $n\ge 2$ 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.