Paul Baird - Böcker

This is the first account in book form of the theory of harmonic morphisms between Riemannian manifolds. Harmonic morphisms are maps which preserve Laplace's equation. They can be characterized as harmonic maps which satisfy an additional first order condition. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with values in a Riemann surface. There are connections with many concepts in differential geometry, for example, Killing fields, geodesics, foliations, Clifford systems, twistor spaces, Hermitian structures, isoparametric mappings, and Einstein metrics, and also the Brownian path-preserving maps of probability theory. Giving a complete account of the fundamental aspects of the subject, this book is self-contained, assuming only a basic knowledge of differential geometry. One chapter follows the complete development of the fundamental geometric aspects of harmonic maps from scratch.This text is suitable for a beginning graduate student interested in harmonic maps and morphisms, or related subjects. The student is brought to the frontiers of knowledge in this rapidly expanding field in which there are many interesting avenues of research to be developed.The authors are world leaders in the field, and have established many of the key results. In this book they have brought together their work and the work of many others to form a coherent account of the subject.This book is the 29th volume in the London Mathematical Society Monographs series, published by Oxford University press on behalf of the London Mathematical Society. The series contains authoritative accounts of current research in mathematics and high quality expository works bringing the reader to the frontiers of research. Of particular interest are topics that have developed rapidly in the past ten years or so, but which have reached a certain level of maturity. Clarity of exposition is important and each book contains preliminary material to make the topic accessible to those commencing work in this area.

This volume has grown from a conference entitled Harmonic Maps, Minimal Sur- faces and Geometric Flows which was held at the Universite de Bretagne Occi- dentale from July 7th-12th, 2002, in the town of Brest in Brittany, France. We welcomed many distinguished mathematicians from around the world and a dy- namic meeting took place, with many fruitful exchanges of ideas. In order to produce a work that would have lasting value to the mathematical community, the organisers decided to invite a small number of participants to write in-depth articles around a common theme. These articles provide a balance between introductory surveys and ones that present the newest results that lie at the frontiers of research. We thank these mathematicians, all experts in their field, for their contributions.Such meetings depend on the support of national organisations and the local community and we would like to thank the following: the Ministere de l'Education Nationale, Ministere des Affaires Etrangeres, Centre National de Recherche Sci en- tifique (CNRS), Conseil Regional de Bretagne, Conseil General du Finistere, Com- munaute Urbaine de Brest, Universite de Bretagne Occidentale (UBO), Faculte des Sciences de l'UBO, Laboratoire de Mathematiques de l'UBO and the Departement de Mathematiques de l'UBO. Their support was generous and ensured the success of the meeting. We would also like to thank the members of the scientific committee for their advice and for their participation in the conception and composition of this volume: Pierre Berard, Jean-Pierre Bourguignon, Frederic Helein, Seiki Nishikawa and Franz Pedit.

This book collects invited contributions by specialists in the domain of elliptic partial differential equations and geometric flows. There are introductory survey articles as well as papers presenting the latest research results. Among the topics covered are blow-up theory for second order elliptic equations; bubbling phenomena in the harmonic map heat flow; applications of scans and fractional power integrands; heat flow for the p-energy functional; Ricci flow and evolution by curvature of networks of curves in the plane.