发表机构
University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究闭定向黎曼4 - 流形上自对偶2 - 形式丛的\(\operatorname{SO}(3)\)联络，利用逐点极分解以正定对称矩阵场\(h\)取代联络，建立相关方程及变分公式等，得出特定形式矩阵场方程等价于反自对偶性和常数量曲率，正Yamabe不变量的反自对偶共形类给出全局解。
AI 中文摘要
对于闭定向黎曼4 - 流形$(M,g)$，我们考虑自对偶2 - 形式丛$\Lambda^+$上的$\operatorname{SO}(3)$联络。在自对偶曲率为保定向框架的开轨迹上，逐点极分解消除规范自由度并用正定对称矩阵场$h$取代联络。我们表明$h$确定唯一兼容联络$A(h)$且杨 - 米尔斯方程等价于二阶系统$\Phi_g(h):=F_{A(h)}^+h^{-1}-g = 0$。我们建立了变分公式、自动不可约性、椭圆正则性以及线性化算子的弗雷德霍姆指标定理。对于形如$h = e^{2\omega}g$的矩阵场，方程$\Phi_g(h)=0$等价于反自对偶性和常数量曲率$6\sqrt{2}$。因此，每个具有正 Yamabe 不变量的反自对偶共形类给出一个全局解。
英文摘要
For a closed oriented Riemannian $4$-manifold $(M,g)$, we consider $\operatorname{SO}(3)$ connections on the bundle $Λ^+$ of self-dual $2$-forms. On the open locus where the self-dual curvature is an orientation-preserving frame, pointwise polar decomposition removes the gauge freedom and replaces the connection by a positive self-adjoint endomorphism field $h$ of $Λ^+$. Using the classical reconstruction of the compatible connection $A(h)$, we obtain a global formulation of the Yang--Mills equation as the determined second order system $$ Φ_g(h):=F_{A(h)}^+h^{-1}-\operatorname{Id}=0. $$ Here, the tensor $F_{A(h)}^+$ is the self-dual curvature of $A(h)$, regarded as an endomorphism of $Λ^+$, and $\operatorname{Id}$ is the identity of $Λ^+$. We establish a variational formulation, automatic irreducibility, elliptic regularity, and a Fredholm index theorem for the linearized operator $D_hΦ_g$. For fields of the form $h=e^{2ω}\operatorname{Id}$, where $ω$ is a smooth real-valued function on $M$, the equation $Φ_g(h)=0$ is equivalent to anti-self-duality and constant scalar curvature $6\sqrt{2}$ of the conformal metric $\hat g=e^{2ω}g$. Consequently, every anti-self-dual conformal class of positive Yamabe constant gives a global solution of $Φ_g(h)=0$.
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