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

TL;DR

This paper analyzes the asymptotic behavior of the gradient flow of self-dual $U(1)$-Higgs energies on manifolds, showing convergence to codimension-two mean curvature flows and enabling the construction of structured Brakke flows from initial cycles.

Contribution

It extends previous stationary results to dynamic flows, demonstrating convergence to mean curvature flows and enabling the creation of structured Brakke flows from initial cycles.

Findings

Solutions converge to codimension-two mean curvature flows as epsilon approaches zero.

The results generalize previous stationary case findings to dynamic gradient flows.

Enables construction of nontrivial Brakke flows starting from given cycles.

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.