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By S. K. Donaldson


theory of Lie groups—but many geometric questions lead to non-standard problems which go beyond, or lie on the frontiers of, standard theory. Such questions arise both in reassuringly down-to-earth topics like the existence of constant mean curvature surfaces in R 3, and in more esoteric ones such as the use of harmonic maps into 'buildings', which have been exploited by Gromov and Schoen to obtain new results on lattices in Lie groups. The main stream in these developments is perhaps the subject of global Riemannian geometry: to discover when a manifold admits metrics with given curvature properties (Einstein metrics, metrics with some positive curvature condition). The Yamabe problem on constant scalar curvature, whose solution was completed by Schoen in 1984, is a conspicuous success in this direction; another notable line of work, principally by Hamilton, studies the evolution of Riemannian metrics by the Ricci tensor flow—a natural non-linear ' q heat ' equation for Riemannian metrics whose fixed points are the Einstein metrics. This has led to concrete new geometric results, for example a characterisation of the 4-sphere through metrics with 'positive curvature operator'. It is interesting that overlapping results have been obtained using a quite different approach—but still relying essentially on analysis and PDE—of Micallef and Moore. They develop the variational theory for harmonic maps of the two-dimensional sphere, and apply it in rather the same way as the variational theory of geodesies is used in more classical pinching theorems. All of these developments are outlined in Schoen's article which gives, in addition to an overall view of progress in the field, a good deal of technical discussion: for example, a new version of results of Helein on regularity for harmonic maps. In sum, this volume contains extremely valuable surveys in a number of the most active areas of research, written by leading authorities, which will have a lot to interest specialists and more general readers. The book is very reasonably priced, and it can be heartily recommended to mathematicians with an interest in differential geometry, interpreted in the broadest terms, and to any research library

Year: 2009
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