The Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows

We develop the asymptotic analysis as ε→0 for the natural gradient flow of the self-dual U(1)-Higgs energies

$$ E_{\varepsilon }(u,\nabla )=\int _{M}\left (|\nabla u|^{2}+ \varepsilon ^{2}|F_{\nabla }|^{2}+ \frac{(1-|u|^{2})^{2}}{4\varepsilon ^{2}}\right ) $$

on Hermitian line bundles over closed manifolds (Mⁿ,g) of dimension n≥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 in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (n−2)-cycle Γ₀ in M, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the (n−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ₀ with additional structure, similar to those produced via Ilmanen’s elliptic regularization.