高维Yang-Mills场的一个奇异性去除定理
A singularity removal theorem for Yang-Mills fields in higher dimensions
摘要：
在四维及更高维空间中，我们证明了如果曲率的$L^2$范数足够小，任何平稳的可容许杨-米尔斯场都可以规范变换为光滑场.有三个主要成分。第一个是普赖斯的单调性公式，它使我们能够断言，不仅在$L^2$范数下，而且在Morrey范数$M_2^{n/2}$下，曲率都很小。第二个成分是一个新的归纳（平均径向）规范建设和截断参数，使我们能够近似一个奇异的规范作为一个弱限制的光滑规范与曲率小的Morrey规范。第二个要素是乌伦贝克引理的变体，允许在曲率的莫雷范数很小的时候，将一个光滑的连接放入库仑规范中;这个变体也由迈耶和里维耶尔独立证明。从这个变式可以很容易地得出，如果$W^{1,2}$ -联络可以用曲率具有小Morrey范数的光滑联络来近似，那么它就可以放在库仑规范中。
英文摘要：
In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small not only in the $L^2$ norm, but also in the Morrey norm $M_2^{n/2}$. The second ingredient is a new inductive (averaged radial) gauge construction and truncation argument which allows us to approximate a singular gauge as a weak limit of smooth gauges with curvature small in the Morrey norm. The second ingredient is variant of Uhlenbeck's lemma, allowing one to place a smooth connection into the Coulomb gauge whenever the Morrey norm of the curvature is small; This variant was also proved independently by Meyer and Riviere. It follows easily from this variant that a $W^{1,2}$-connection can be placed in the Coulomb gauge if it can be approximated by smooth connections whose curvatures have small Morrey norm.
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