| Summary: | We study different properties of harmonic maps between two compact Riemannian manifolds M and M' and in particular the following existence question: do there exist harmonic elements in the different homotopy classes of maps from M to M'? Our first result is an affirmative answer to that question when M is a surface and the second homotopy group of M' is zero. This result is then extended to certain products of manifolds and to ψ-harmonic maps. When M and M' are orientable surfaces, this solves the existence question as long as M' is not a sphere. When M is a surface of genus p and M' a sphere, the question was answered by J. Eells and J. Wood for all classes of maps of degree greater or equal to p. For all remaining cases (degree ≤ p - 1), we obtain existence results for particular metrics on M and M'. We also study the question of existence for non-orientable surfaces, and obtain complete results for maps between spheres and projective planes. A second type of results is a finiteness theorem for harmonic maps: we show that if the sectional curvature of M' is negative, there is only a finite number of non-constant harmonic maps from M to M' of dilatation bounded by a fixed constant. This implies a similar result on the number of almost complex maps between almost Kaehlerian manifolds. Other results include an example of a continuous family of harmonic maps, a remark on the derivatives of the harmonicity equations and an answer to a question of H. Eliasson concerning a higher order energy.
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