The global existence of Yang-Mills fields on curved space-times

Abstract

This is an introductory chapter in a series in which we take a systematic study of the Yang-Mills equations on curved space-times. In this first, we provide standard material that consists in writing the proof of the global existence of Yang-Mills fields on arbitrary curved space-times using the Klainerman-Rodnianski parametrix combined with suitable Grönwall type inequalities. While the Chruściel-Shatah argument requires a simultaneous control of the $L_{loc}^{\infty }$ and the $H_{loc}^2$ norms of the Yang-Mills curvature, we can get away by controlling only the $H_{loc}^1$ norm instead, and write a new gauge independent proof on arbitrary, fixed, sufficiently smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds. This manuscript is written in an expository way in order to provide notes to Master's level students willing to learn mathematical General Relativity.

arXiv comment75 pages

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