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Title: On the existence of harmonic maps
Author: Lemaire, Luc
ISNI:       0000 0001 3608 0282
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1977
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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.
Supervisor: Not available Sponsor: Fonds national de la recherche scientifique (Belgium) ; Royal Society ; British Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics