The parabolic U(1)-Higgs equations and codimension-two mean curvature flows

We develop the asymptotic analysis as the scaling parameter tends to zero for the natural gradient flow of the self-dual U(1)-Higgs energies on Hermitian line bundles over closed manifolds of dimension at least 3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows, i.e., integral (n−2)-Brakke flows, generalizing results of Pigati and Stern from the stationary case. Given any integral (n−2)-cycle in M, these results can be used together with the convergence theory developed by the authors to produce nontrivial integral Brakke flows starting at the given cycle with additional structure, similar to those produced via Ilmanen’s elliptic regularization.