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

Abstract

We develop the asymptotic analysis as $\epsilon \to 0$ for the natural gradient flow of the self-dual $U(1)$-Higgs energies $$E_{\epsilon }(u,\nabla )={\int }_M\left(|\nabla u{|}^2+{\epsilon }^2|F_{\nabla }{|}^2+\frac{(1-|u{|}^2{)}^2}{4{\epsilon }^2}\right)$$ on Hermitian line bundles over closed manifolds $(M^n,g)$ of dimension $n\ge 3$, 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 the last two authors from the stationary case. Given any integral $(n-2)$-cycle ${\Gamma }_0$ in $M$, these results can be used together with the convergence theory developed in previous work of the authors to produce nontrivial integral Brakke flows starting at ${\Gamma }_0$ with additional structure, similar to those produced via Ilmanen's elliptic regularization.

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