Uhlenbeck Compactness
Uhlenbeck Compactness
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DOI：
10.4171/004
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通讯作者：
K. Wehrheim
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作者：
K. Wehrheim
前言 本书的起源在于我刚开始研究生学习时，当时我还无法理解 Uhlenbeck 紧性，更不用说了解它是否也适用于我的案例——有边界的流形。标准数学教育与理解杨-米尔斯理论中非常基础的研究论文所需的分析背景之间似乎存在一定的差距。当我与 Laurent Lazzarini 合作时，第一个差距被缩小——我们终于理解了算子 d ⊕ d * 的 L p 估计。从那时起，在我的主管 Dietmar Salamon 的指导和鼓励下，我继续深入 Uhlenbeck 紧凑性，并始终牢记我的界限。我必须克服一些障碍，发现一些微妙之处，并最终详细了解 Uhlenbeck 紧性及其对具有边界的流形和紧集耗尽流形的（离明显不远的）概括。所有这些工作似乎都值得写下来，所以我最终写了一本我在研究生学习开始时需要的书：对乌伦贝克紧性的独立阐述以及所有的分析细节，它只引用标准教科书的经典结果。在难以找到有关诺依曼边值问题的 L p 结果的参考资料后，我包含了有关该主题的初步部分。这为诺依曼问题提供了所需的结果，也展示了椭圆边值问题背后的一般哲学——它教会了我乌伦贝克的句子“椭圆系统在索博列夫空间上表现良好”中的深层真理。因此，本书旨在为学生分析杨-米尔斯理论提供指导。我也希望这对于那些需要了解 Uhlenbeck 紧致性中或背后的特定细节的人来说是一个有用的参考。让我强调一下，我不主张任何原创作品。本书中阐述的 Uhlenbeck 紧性的小概括以前已经为人所知。然而，证明的某些部分有很多细节和替代方法，这些细节和替代方法可能在其他地方找不到。这些细节在引言中都有详细说明。这本书的大部分功劳都归功于迪特马尔·萨拉蒙（Dietmar Salamon）——他提出了这个想法，帮助我克服障碍，还教我如何自己克服这些障碍，最后教我如何写数学。 ……
Preface The origin of this book lies at the beginning of my graduate studies, when I just could not understand Uhlenbeck compactness, let alone see whether it would also hold for my cases – on manifolds with boundary. There seemed to be certain gaps between standard mathematics education and the analytic background needed to understand a very fundamental research paper in Yang-Mills theory. A first gap was closed while I was working with Laurent Lazzarini – we finally understood the L p-estimate for the operator d ⊕ d *. From there I went on through the depth of Uhlenbeck compactness-guided and encouraged by my supervisor, Dietmar Salamon, and always keeping my boundaries in mind. I had to overcome some obstacles, found a few subtleties, and finally arrived at a detailed understanding of Uhlenbeck compactness and its (not far from obvious) generalizations to manifolds with boundaries and manifolds exhausted by compact sets. All this work seemed to be worth writing down, so I ended up writing the book that I would have needed at the beginning of my graduate studies: A selfcontained exposition of Uhlenbeck compactness with all the analytic details, which only refers back to standard textbooks for classical results. After having difficulties in finding references on L p-results for the Neumann boundary value problem I included a preliminary part on that topic. This provides the required results for the Neumann problem and also shows the general philosophy behind elliptic boundary value problems – it taught me the deep truth in Uhlenbeck's sentence " Elliptic systems are well-behaved on Sobolev spaces ". So this book intends to be a guide to students on their way into the analysis of Yang-Mills theory. I also hope that it will be a useful reference for those who need to know about a particular detail in or behind Uhlenbeck compactness. Let me stress that I do not claim any original work. The minor generalizations of Uhlenbeck compactness that are stated in this book have been known before. However, there are a lot of details and alternative approaches to certain parts of the proofs that probably cannot be found elsewhere. These bits and pieces are specified in the introduction. Much credit for this book goes to Dietmar Salamon-for the idea as a start, for all the help with obstacles, but also for teaching me how to overcome them myself and finally, how to write mathematics. …