We study the limiting behavior as \(p\uparrow 2\) of the singular sets \(Sing(u_p)\) and p-energy measures \(\mu _p:=(2-p)|du_p|^pdvol\) for families of stationary p-harmonic maps \(u_p\in W^{1,p}(M,S^1)\) from a closed, oriented manifold M to the circle. When the measures \(\mu _p\) have uniformly bounded mass, we show that—up to subsequences—the singular sets \(Sing(u_p)\) converge in the Hausdorff sense to the support of a stationary, rectifiable varifold V of codimension 2, and the measures \(\mu _p\) converge weakly in \((C^0(M))^*\) to a limit of the form

where h is a harmonic one-form. For solutions on two-dimensional domains, we show moreover that the density of V takes values in \(2\pi {\mathbb {N}}\). Finally, we observe that nontrivial families \(u_p\) of such maps arise naturally on any closed Riemannian manifold of dimension \(n\ge 2\), via variational methods.

我们以奇异集\（Sing（u_p）\）和p-能量测度\（\ mu _p：=（2-p）| du_p | ^ pdvol \）的极限行为作为（p \ uparrow 2 \）用于从闭合定向歧管M到圆的平稳p调和映射\（u_p \ in W ^ {1，p}（M，S ^ 1）\）的族。当测度\（\ mu _p \）具有一致的边界质量时，我们证明，直到子序列为止，奇异集\（Sing（u_p）\）在Hausdorff方向上收敛，得到了一个固定的，可校正的V变量V的支持。余维2，并且测度\（\ mu _p \）在\（（C ^ 0（M））^ * \）中弱收敛 到形式的极限

其中h是谐波单形式。对于二维域上的解，我们进一步证明V的密度取\（2 \ pi {\ mathbb {N}} \）中的值。最后，我们观察到，这些图的非平凡族\（u_p \）通过变分方法自然出现在任何维\（n \ ge 2 \）的闭合黎曼流形上。