The Ricci flow: techniques and applications -Part I: Geometric aspects (Mathematical Surveys and Monographs 135)
The Ricci flow: techniques and applications -Part I: Geometric aspects (Mathematical Surveys and Monographs 135)
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DOI：
10.1112/blms/bdm088
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发表时间：
2008-02
影响因子：
0.9
通讯作者：
P. Baird
中科院分区：
文献类型：
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作者：
P. Baird
近年来在理解几何与低维拓扑之间的关系方面取得了显著的进展。R.汉密尔顿，谁率先Ricci流作为一种技术来解决瑟斯顿的几何化猜想，以及最近的成就，G。佩雷尔曼在这方面的计划，构成壮观的例子。这些成功可能在这一领域的专家中注入了一种“失望”的感觉：取得了这么多成就，我现在能做什么？然而，如果是这样的话，这种情绪是完全没有道理的。因为，而像物理学的世界，每一个发现似乎发现一个新的袋子的技巧，所以工作的汉密尔顿，佩雷尔曼和其他人揭示了一个全新的世界的想法和问题，丰富在许多方面比一个人可以想象的。考虑到这一点，这本书的里奇流：技术和应用.由B。Chow等人，与早期的卷[1]，是一个宝石。如果我们在做数学，而不是被告知一个整齐缝合的主题，没有什么可做的，我们希望有挑战性的问题要解决，有人以连贯和动手的方式向我们解释这些问题，这将使我们能够深入了解它们。这正是这本书所做的，因此我认为它对研究里奇流和密切相关领域的专家来说是无价的，如平均曲率流，低维几何和拓扑学，以及更一般的几何分析。
The progress made in recent years in understanding the relation between geometry and lowdimensional topology has been remarkable. The insights of R. Hamilton, who has pioneered the Ricci flow as a technique to tackle Thurston’s geometrisation conjecture, together with the recent achievements of G. Perelman in respect of this program, constitute spectacular examples. These successes may have imbued a sense of ‘let down’amongst specialists working in this area: with so much achieved, what can I do now? However, if this were to be the case, such a sentiment would be completely unjustified. For, rather like the world of physics, where each discovery seems to uncover a new bag of tricks, so the work of Hamilton, Perelman and others has revealed a whole new world of ideas and problems, richer in many ways than one could have imagined.With this in mind, the book The Ricci flow: techniques and applications... by B. Chow et al., together with the earlier volume [1], is a gem. If we are in the business of doing mathematics, rather than being told about a neatly sewn-up subject where there is nothing left to do, we want challenging problems to solve and someone to explain these problems to us in a coherent and hands-on way that will allow us to get our teeth into them. This is precisely what this book does, and as such I would judge it to be invaluable to specialists working on the Ricci flow and closely related areas, such as mean-curvature flow, low-dimensional geometry and topology and, more generally, geometric analysis.