arXiv 2103.146152021-03-26math.DGmath.AP

self-dual $U(1)$ -Yang-Mills-Higgs能量向$(n-2)$ -面积泛函的收敛

Convergence of the self-dual $U(1)$-Yang-Mills-Higgs energies to the $(n-2)$-area functional

Davide Parise, Alessandro Pigati, Daniel Stern

备注：: 52 pages; comments welcome

摘要：

给定闭黎曼流形$(M^n,g)$上的厄米线丛$L\to M$，自对偶杨-米尔斯-希格斯能量是定义在由截面$u\inΓ(L)$和具有曲率$F_\nabla$的厄米联络$\nabla$组成的偶$(u,\nabla)$上的自然泛函族\begin{align*} &E_ε(u,\nabla):=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4ε^2}\Big) \end{align*}.虽然这些泛函的临界点在二维中已经被规范理论界研究得很好，但在第二和第三作者之前的工作中表明，在高维中的临界点以PDMATHAMT（在适当的意义上）收敛到余维2的极小子流形，与艾伦-卡恩方程和极小超曲面之间的对应关系有很强的相似之处。在本文中，我们补充这一想法，显示$Γ$ -收敛$E_ε$（$2π$倍）的余维两个区域：更准确地说，给出一个家庭的夫妇$(u_ε,\nabla_ε)$与$\sup_εE_ε(u_ε,\nabla_ε)<\infty$，我们证明了一个合适的规范不变雅可比$J(u_ε,\nabla_ε)$收敛到一个整体$(n-2)$ -周期$Γ$，在同调类对偶的欧拉类$c_1(L)$，质量$2 π\mathbb{M}(Γ)\le\liminf_{ε\rightarrow 0}E_ε(u_ε,\nabla_ε)$。我们还得到了该同调类中任意整圈的恢复序列。最后，我们应用这些技术来比较最小-最大值的$(n-2)$ -面积从Almante-Pitts理论与那些从杨米尔斯希格斯框架，表明前者的值总是为后者提供了一个下限。作为一个组成部分，我们还建立了一个Huisken型单调性结果沿$E_ε$的梯度流。

英文摘要：

Given a hermitian line bundle $L\to M$ on a closed Riemannian manifold $(M^n,g)$, the self-dual Yang-Mills-Higgs energies are a natural family of functionals \begin{align*} &E_ε(u,\nabla):=\int_M\Big(|\nabla u|^2+ε^2|F_\nabla|^2+\frac{(1-|u|^2)^2}{4ε^2}\Big) \end{align*} defined for couples $(u,\nabla)$ consisting of a section $u\inΓ(L)$ and a hermitian connection $\nabla$ with curvature $F_\nabla$. While the critical points of these functionals have been well-studied in dimension two by the gauge theory community, it was shown in previous work of the second- and third-named authors that critical points in higher dimension converge as $ε\to 0$ (in an appropriate sense) to minimal submanifolds of codimension two, with strong parallels to the correspondence between the Allen-Cahn equations and minimal hypersurfaces. In this paper, we complement this idea by showing the $Γ$-convergence of $E_ε$ to ($2π$ times) the codimension two area: more precisely, given a family of couples $(u_ε,\nabla_ε)$ with $\sup_εE_ε(u_ε,\nabla_ε)<\infty$, we prove that a suitable gauge invariant Jacobian $J(u_ε,\nabla_ε)$ converges to an integral $(n-2)$-cycle $Γ$, in the homology class dual to the Euler class $c_1(L)$, with mass $2 π\mathbb{M}(Γ)\le\liminf_{ε\rightarrow 0}E_ε(u_ε,\nabla_ε)$. We also obtain a recovery sequence for any integral cycle in this homology class. Finally, we apply these techniques to compare min-max values for the $(n-2)$-area from the Almgren-Pitts theory with those obtained from the Yang-Mills-Higgs framework, showing that the former values always provide a lower bound for the latter. As an ingredient, we also establish a Huisken-type monotonicity result along the gradient flow of $E_ε$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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