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

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS

ACS Applied Bio Materials Pub Date : 2024-05-29 DOI:10.1007/s00039-024-00684-9

Davide Parise, Alessandro Pigati, Daniel Stern

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Abstract

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.

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抛物线 U(1)-Higgs 方程与二维平均曲率流

我们对自双 U(1)-Higgs 能量的自然梯度流 $$ E_{\varepsilon }(u.)进行了 ε→0 的渐近分析、\nabla )=\int _{M}\left (|\nabla u|^{2}+ \varepsilon ^{2}|F_{\nabla }|^{2}+ \frac{(1-|u|^{2})^{2}}{4\varepsilon ^{2}}\right ) $$ 在封闭流形（Mn、g) 上的赫米线束上的 $$，表明解在度量理论意义上收敛于编码维数为 2 的平均曲率流--即.e.,223:1027-1095, 2021）的结果。给定 M 中的任何积分（n-2）循环Γ0，这些结果可以与（Parise 等人在《自双 U(1)-Yang-Mills-Higgs 能量向（n-2）面积函数的收敛》中，2021 年，arXiv:2103.14615）中发展的收敛理论一起使用，以产生从Γ0 开始的具有额外结构的非难积分布拉克流，类似于通过伊尔马宁的椭圆正则化产生的布拉克流。

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来源期刊

ACS Applied Bio Materials Chemistry-Chemistry (all)

CiteScore

9.40

自引率

2.10%

发文量

464