Burstall, Francis E. (1984) Non-linear functional analysis and harmonic maps. PhD thesis, University of Warwick.

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Abstract

Harmonic maps are the solutions of a natural variational problem in Differential Geometry. This thesis is concerned with questions of existence, classification and special properties of harmonic maps.

1. Existence:
Variational arguments are used to establish the existence of harmonic maps of finite energy from non-compact manifolds when either

(a) the target manifold is compact and saosfies certain geometrical conditions, or
(b) the domain is two-dimensional and the target satisfies certain growth conditions.

Further, infinite-dimensional differentiable structures are exhibited for certain spaces of maps that arise naturally in this context.

2. Classification:
The twistorial methods of Eells-Salamon and Rawnsley are exploited to classify strongly conformal harmonic maps of a Riemann surface into a Grassmannian by holomorphic maps of the surface into a flag manifold equipped with a special non-integrable almost complex structure.

Similar ideas are used to classify isotropic harmonic maps of a Riemann surface into a space form by f-holomorphic maps into bundles of f-structures over the space form.

In this context, we also examine the relevant properties of f-structures and f-holomorphic maps and, in particular, show the existence of a homotopy invariant for maps of cosymplectic manifolds into f-Kahler manifolds generalising that of Lichnerowicz.

3. Properties:
A characterisation in terms of harmonic maps of those maps between Riemannian manifolds that commute with the co-differential is given.

Unique continuation properties of harmonic maps are considered and in the case of two-dimensional domains, proved by use of holomorphic differentials. In particular, we establish unique continuation of isotropy for branched minimal surfaces in a space form.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Nonlinear functional analysis, Harmonic maps
Official Date:	June 1984
Dates:	Date Event June 1984 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Eells, James, 1926-2007
Extent:	xv, 118 leaves
Language:	eng

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