Non-linear functional analysis and harmonic maps
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.