Some Results On Harmonic Maps And Pseudoharmonic Maps

Harmonic maps are one of the most important topics of differential geometry, which has been widely applied to geometry topology and theoretical physics. Besides the harmon-ic map, we also study some generalized harmonic maps which are called pseudoharmonic maps.The thesis is divided into five chapters.In chapter1, we introduce the background and current situation on the subject. After that, the main results in this thesis are presented.In chapter2, we mainly study the geometrical structures of CR manifolds and some analytic properties of harmonic maps and pseudoharmonic maps. First, some preliminar-ies on CR manifolds are introduced, including the relationship between pseudoharmonic maps and harmonic maps and the properties of CR pluriharmonic maps. Second, the general CR Paneitz operator is used to get the Bochner-type Theorem. Then by moving frame method and the Sampson technique, we prove the CR pluriharmonicity of harmonic maps and pesudoharmonic maps, especially under suitable rank conditions these maps are CR holomorphic. Finally we find that, under some energy decay conditions, the har-monic maps and pseudoharmonic maps from complete noncompact CR manifold are CR pluriharmonic.In chapter3, the properties of the harmonic maps from CR manifolds to locally symmetric spaces are discussed. First we give the relationship between the Siu-Sampson results of CR type and the Lie algebra, generalizing the results of Carlson and Toledo. If the target manifold is locally Hermitian symmetric space of noncompact type, under suitable rank condition, we get the CR holomorphicity of the harmonic maps. Then we discuss the basic stability of the harmonic maps. Under certain rank condition, the basic stable CR plurihamonic map is CR holomorphic when target manifold is irreducible Hermitian symmetric space of compact type.Chapter4is devoted to the Hermitian harmonic maps from Hermitian manifold into Kahler manifold. Assuming the domain manifolds posses some special exhaustion func-tions and the vector field V=JM δJM satisfies some decay conditions, some monotonic-ity formulas of partial energies of Hermitian pluriharmonic maps are established. These monotonicity inequalities enable us to derive the holomorphicity for these Hermitian pluri-harmonic maps.In chapter5, the harmonic maps with potential are considered. We obtain the mono-tonicity formulae and Liouville theorems for the harmonic maps with potential under some conditions on the potential. We also obtain the unique constant solution of the constant Dirichlet boundary value problem on some starlike domain for harmonic maps with potential.