超临界维数中弱$G$-丛的拓扑与逼近
Topology and approximation of weak $G$-bundles in the supercritical dimensions
AI总结：
本文为超临界维数中Morrey-Sobolev主$G$-丛-联络对引入拓扑同构类，并证明其可逼近性，结果类比流形值VMO映射的拓扑度理论。
摘要：
为了分析高维中的驻定杨-米尔斯联络，必须处理Morrey-Sobolev丛和联络。Morrey-Sobolev主$G$-丛的转移映射不连续，因此通常的拓扑概念没有意义。在本工作中，我们为丛-联络对$\left( P, A\right)$发展了拓扑同构类的概念，并利用这些概念在Morrey-Sobolev框架下推导出丛和联络的若干可逼近性结果。我们的证明采用面向联络的方法，并强调了在低正则性区域中，丛和联络的正则性是相互交织的。我们的结果与流形值$\mathrm{VMO}$映射的拓扑度理论和逼近结果相类似。
英文摘要：
For analyzing stationary Yang-Mills connections in higher dimensions, one has to work with Morrey-Sobolev bundles and connections. The transition maps for a Morrey-Sobolev principal $G$-bundles are not continuous and thus the usual notion of topology does not make sense. In this work, we develop the notion of a topological isomorphism class for a bundle-connection pair $\left( P, A\right)$ and use these notions to derive several approximability results for bundles and connections in the Morrey-Sobolev setting. Our proofs follow a connection-oriented approach and also highlight the fact that in the low regularity regime, the regularity of the bundle and connection are intertwined. Our results parallels the theory of the topological degree and approximation results for manifold-valued $\mathrm{VMO}$ maps.
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