Lemaire, Luc (1977) On the existence of harmonic maps. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Harmonic maps, Riemannian manifolds, Homotopy groups
Official Date:	May 1977
Dates:	Date Event May 1977 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Eells, James, 1926-2007
Sponsors:	Fonds national de la recherche scientifique (Belgium) ; Royal Society (Great Britain) ; British Council
Format of File:	pdf
Extent:	iv, 69 leaves : illustrations
Language:	eng

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