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Harmonic maps between surfaces (with a special chapter on conformal mappings). (English) Zbl 0542.58002
Lecture Notes in Mathematics. 1062. Berlin etc.: Springer-Verlag. X, 133 p. DM 19.80; $ 7.20 (1984).
The primary objective of this excellent monograph is to give a detailed account of what is known at present about harmonic maps between surfaces and to give a fairly self-contained introduction to the general harmonic map theory. Among the topics treated in the first part of the book, the author gives a detailed proof of the classical Morrey’s theorem on conformal maps, solves the Dirichlet problem for boundary values lying in a convex ball [H. Kaul, S. Hildebrandt and K. Widman, Acta Math. 138, 1-16 (1977 Zbl 0356.53015)] and presents a new proof of Lemaire’s existence theorem as well as the uniqueness theorems for harmonic maps due to Hartman and Jäger-Kaul. The rest of the book is mainly devoted to \(C^{1,\alpha}\)-a-priori estimates yielding e.g. existence of harmonic diffeomorphisms (as solutions of the Dirichlet problem). Harmonic coordinates due to Jost-Karcher are introduced to prove \(C^{2,\alpha}\)-a-priori estimates for harmonic maps between surfaces; these are then exploited to prove existence of harmonic diffeomorphisms between closed surfaces [the author and R. Schoen, Inv. Math. 66, 353-359 (1982; Zbl 0488.58009)]. Finally, the author presents Eells-Wood’s analytic proof of a result of Kneser on mappings between closed surfaces together with some applications of harmonic maps to Teichmüller theory.
Reviewer: G.Tóth

MSC:
58E20 Harmonic maps, etc.
58J32 Boundary value problems on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis