Harmonic maps from surfaces to complex projective spaces and certain Lie groups

In this thesis we are concerned with harmonic maps from a Riemann surface to a complex projective space, the unitary group, the orthogonal group, or the symplectic group.

We describe and link two constructions of complex isotropic (equivalently, finite uniton number) harmonic maps from a Riemann surface to complex projective spaces; all harmonic maps from the 2-sphere are complex isotropic. We then specialise to harmonic maps from the 2-sphere to the complex projective plane and show that there is no restriction on the ramification behaviour in some situations and that the opposite is true in other situations.

We find the dimension of the spaces of holomorphic sections and holomorphic differentials of certain line bundles. We use those results to give improved lower bounds on the index of complex isotropic harmonic maps from the 2-sphere and torus to a complex projective space of arbitrary dimension and from higher genus surfaces in some cases.

We give, up to dimension 6, algebraic parametrizations of all S^1-invariant extended solutions of harmonic maps of finite uniton number from a Riemann surface to the symplectic group, giving the corresponding harmonic maps explicitly. For arbitrary dimension we give an algorithm which parametrizes all such S^1-invariant extended solutions of harmonic maps which are of standard type, i.e., of the maximum possible uniton number.