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The role of harmonic mappings in rigidity and deformation problems. (English) Zbl 0806.58013
Komatsu, Gen (ed.) et al., Complex geometry. Proceedings of the Osaka international conference, held in Osaka, Japan, Dec. 13-18, 1990. New York: Marcel Dekker. Lect. Notes Pure Appl. Math. 143, 179-200 (1993).

The present paper discusses various aspects of harmonic maps as they enter in rigidity and deformation problems. In Section 1 the author concentrates on convexity properties of the energy functional for maps into nonpositively curved manifolds and gives an example for the failure of convexity in the positivity case. The core material is in Section 2 which contains many results of the theory of harmonic diffeomorphisms of surfaces. The author proves here a Bernstein-type theorem for minimal maps and existence and uniqueness of area preserving maps between compact hyperbolic surfaces. Several conjectures about harmonic diffeomorphisms are also included.

In the final section the author discusses some applications of harmonic maps to rigidity problems for discrete groups; a joint work of the author and M. Gromov.

Reviewer: G.Tóth (Camden)

MSC:
58E20Harmonic maps between infinite-dimensional spaces