Title:

Some properties of harmonic mappings

A harmonic map between Riemannian manifolds satisfies, in local coordinates, a second order semilinear elliptic system of equations. This system of equations arise as the EulerLagrange equations of a natural Dirichlet or energy integral on maps between manifolds, which directly generalises the classical Dirichlet integral. Particular interest in harmonic maps has thus sprung up in connection with the problem of minimal surfaces in Riemannian manifolds. This thesis begins with a brief introduction to harmonic maps, putting the concepts into a general framework and recording certain basicbut important properties of harmonic maps. The second chapter is founded on a remark of Eells and Sampson [8] that a holomorphic map between KWhler manifolds is harmonic. Here a calculation is made of the Laplacian of a decomposed energy density and application of it is made in the holomorphic case. The formula obtained is used in conjunction with harmonic map methods to give a greatly simplified proof of a theorem of Cheng characterising the Euclidean sphere by the eigenfunctions of its Laplacian. Up until the beginning of the writing of this thesis hardly anything was known about harmonic maps from noncompact domain. Chapter three deals with two situations, one ensuring that the energy density is bounded and another ensuring the total energy is infinite, some contrasts are given including a counterexample to a tempting conjecture. While some of these results rely on curvature restrictions a separate chapter has been reserved for this topic and among those areas considered are maps from manifolds with boundary, a classification problem for maps of small energy and a few brief remarks about the second variation. Chapter five is a discussion of an old paper of Shibata concerning the existence of harmonic diffeomorphisms of surfaces in which many mistakes have been found. Many of these are corrected but the final solution is not found and an alternative approach to the problem is proposed. A short appendix is attached in which the connection between certain harmonic and holomorphic maps is pointed out. This is viewed as a special case in which an equidistribution theory for harmonic maps actually exists, nothing of this nature is known in general.
