For any compact Lie group 𝐺 and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\ge 2$, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with $L^p$-small curvature, when $p>d/2$, to the case of a connection with $L^{d/2}$-small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension $d\ge 2$, when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

中文翻译：

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最优Łjajaiewicz–Simon不等式和Morse–Bott Yang–Mills能量函数

对于任何紧凑的Lie组𝐺和闭合的光滑黎曼流形 $（X，G）$ 尺寸 $d\ge 2$，我们扩展了Uhlenbeck（1985）的结果，该结果给出了主𝐺束上平坦的连接的存在，而𝐺支撑了与 ${\mathrm{大号}}^p$-小曲率，当 $p>d/2$，如果与 ${\mathrm{大号}}^{d/2}$-小曲率。我们证明了Banach流形上抽象Morse-Bott函数的最优Łojasiewicz-Simon梯度不等式，归纳了作者和Maridakis（2019）提出的较早结果，主要是通过消除了Hessian算子是索引为零的Fredholm的假设。我们用这个结果证明最优的Łojasiewicz-Simon梯度不等式对于闭合黎曼四流形上规则反自对偶连接附近的自对偶Yang-Mills能量函数以及在闭合黎曼流形上的完整Yang-Mills能量函数尺寸$d\ge 2$，在给定的Yang-Mills关系中被称为Morse-Bott。我们还通过在给定的平面连接（即曲率图的规则点）附近进行直接分析，证明了最优Łojasiewicz–Simon梯度不等式。