Preface to the Fourth Edition
Almost twenty years after conception of the first edition, it was a challenge to prepare an updated version of this text on the Calculus of Variations. The field has truely advanced dramatically since that time, to an extent that I find it impossible to give a comprehensive account of all the many important developments that have occurred since the last edition appeared. Fortunately, an excellent overview of the most significant results, with a focus on functional analytic and Morse theoretical aspects of the Calculus of Variations, can be found in the recent survey paper by Ekeland-Ghoussoub. I therefore have only added new material directly related to the themes originally covered.
Even with this restriction, a selection had to be made. In view of the fact that flow methods are merging as the natural tool for studying variational problems in the field of Geometric Analysis, an emphasis was placed on advances in this domain. In particular, the present edition includes the proof for the convergence of the Yamabe flow on an arbitrary closed manifold of dimension 3 ≤ m ≤ 5 for initial data allowing at most single-point blow-up. Moreover, we give a detailed treatment of the phenomenon of blow-up and discuss the newly discovered results for backward bubbling in the heat flow for harmonic maps of surfaces.
Aside from these more significant additions, a number of smaller changes have been made throughout the text, thereby taking care not to spoil the freshness of the original presentation. References have been updated, whenever possible, and several mistakes that had survived the past revisions have now been eliminated. I would like to thank Silvia Cingolani, Irene Fonseca, Emmanuel Hebey, and Maximilian Schultz for helpful comments in this regard. Moreover, I am indebted to Gilles Angelsberg, Ruben Jakob, Reto Muller, and Melanie Rupflin, for carefully proof-reading the new material.
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