Geometric and Functional Analysis ( IF 2.4 ) Pub Date : 2024-05-29 , DOI: 10.1007/s00039-024-00684-9 Davide Parise , Alessandro Pigati , Daniel Stern
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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.
中文翻译：
抛物线U(1)-希格斯方程和余维-双平均曲率流
我们将自对偶 U(1)-希格斯能量的自然梯度流的渐近分析发展为 ε→0
$$ E_{\varepsilon }(u,\nabla )=\int _{M}\left (|\nabla u|^{2}+ \varepsilon ^{2}|F_{\nabla }|^{2} + \frac{(1-|u|^{2})^{2}}{4\varepsilon ^{2}}\right ) $$
在维度 n≥3 的闭流形 (M n ,g) 上的埃尔米特线束上，表明解在测度理论意义上收敛于余维二平均曲率流，即积分 (n−2 )-Brakke 流——（Pigati 和 Stern in Invent. Math. 223:1027–1095, 2021）在固定情况下的概括结果。给定 M 中的任何积分 (n−2)-循环 Γ 0 ，这些结果可以与 (Parise 等人在自对偶 U(1)- 的收敛性中) 中开发的收敛理论一起使用Yang–Mills–Higgs 能量到 (n−2) 区域泛函，2021，arXiv:2103.14615)，以产生从 Γ 0 开始具有附加结构的非平凡积分 Brakke 流，类似于通过 Ilmanen 椭圆产生的流正则化。
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