A kinetic characterization of vortices

Motivated by the study of a micromagnetic enery, Jabin, Otto and Perthame proved in a 2002's paper that in dimension 2, we can select curl-free unit-lenght vector field through a kinetic formulation. Roughly speaking, this kinetic formulation excludes curl-free unit-lenght vector field which are singular along $\mathcal{H}^1$-set to keep only solutions with point-like singularities. We will give a natural generalization to this kinetic formulation in any dimension and show that if the dimension is strictly greater than 2, it becomes much more rigid as it characterizes constant vector field or vortices, that is $\pm\frac{x-P}{|x-P|}$; if we have enough time, we will show how this is related with totally umbilical surfaces.