Univ., Diss.--Sendai, 1999.

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by Hung Ching Nam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaf 100). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Sobolev spaces of maps into metric space --- p.9 / Chapter 2.1 --- "Lp(Ω,X) spaces" --- p.9 / Chapter 2.2 --- Maps with finite energy --- p.11 / Chapter 2.3 --- Differentiation of maps along a direction --- p.28 / Chapter 2.4 --- Theory of differentiation of maps --- p.35 / Chapter 2.5 --- Trace of maps on Lipschitz domains --- p.48 / Chapter 3 --- Sobolev maps into NPC space --- p.58 / Chapter 3.1 --- NPC space --- p.58 / Chapter 3.2 --- NPC space with curvature bound --- p.69 / Chapter 3.3 --- Sobolev maps into NPC space --- p.71 / Chapter 3.4 --- Tensor inequality for Sobolev maps --- p.77 / Chapter 4 --- Harmonic maps into NPC space --- p.79 / Chapter 4.1 --- Existence and uniqueness of Dirichlet problem --- p.79 / Chapter 4.2 --- Interior Lipschitz continuity of harmonic maps --- p.81 / Chapter 5 --- Equivariant harmonic maps --- p.86 / Chapter 5.1 --- A functional analysis lemma --- p.86 / Chapter 5.2 --- Existence of equivariant harmonic maps --- p.87 / Chapter 5.3 --- Compactification of NPC space --- p.93 / Chapter 5.4 --- Isometric action on CAT(-l) space --- p.96

by Tsui Wai-kwok Ricky. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 58-59). / Abstracts in English and Chinese. / Chapter 1 --- Preliminary --- p.4 / Chapter 1.1 --- Introduction --- p.4 / Chapter 1.2 --- Some basic theorem --- p.7 / Chapter 2 --- Bubble tree Convergence for a sequence of harmonic map --- p.11 / Chapter 3 --- Heat Flow of Harmonic Maps on Riemann Surface --- p.21 / Chapter 3.1 --- Existence of unique solution to the evolution problem --- p.21 / Chapter 3.1.1 --- Some Basic Estimates --- p.22 / Chapter 3.1.2 --- Existence Result --- p.34 / Chapter 3.1.3 --- Behaviour of solutions near singular points --- p.37 / Chapter 3.2 --- Finite time Blow-up --- p.39 / Chapter 3.3 --- Energy Identity --- p.51 / Bibliography --- p.58

Chan Mei Shan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 74-76). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.1 / Chapter 1 --- Preliminary --- p.9 / Chapter 1.1 --- Harmonic maps between Riemann Surfaces --- p.9 / Chapter 1.2 --- "Minkowski 3-spaces, M21" --- p.17 / Chapter 1.3 --- Preliminaries from analysis --- p.21 / Chapter 2 --- Holomorphic quadratic differentials --- p.27 / Chapter 2.1 --- Solution on the Poincare disk D --- p.28 / Chapter 2.2 --- Solution on the complex plane C --- p.37 / Chapter 3 --- Harmonic Diffeomorphisms into H2 --- p.46 / Chapter 3.1 --- The case from D onto D --- p.46 / Chapter 3.2 --- Open harmonic embeddings from C into H --- p.53 / Chapter 4 --- Open harmonic embeddings with polynomial Hopf differentials --- p.57 / Chapter 4.1 --- Proof of the theorem --- p.58 / Chapter 4.2 --- Open harmonic embeddings on C with fixed ideal polygonal images in H2 --- p.65 / Bibliography --- p.74

by Lam Kwan Hang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 75-76). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Maps into locally compact Riemannian complex --- p.8 / Chapter 2.1 --- Exitsence of energy minimizing maps --- p.8 / Chapter 2.2 --- Length minimizing curves --- p.10 / Chapter 3 --- Harmonic maps into nonpositively curved spaces --- p.13 / Chapter 3.1 --- Nonpositively curved spaces --- p.13 / Chapter 3.2 --- Properties of the distance function --- p.16 / Chapter 4 --- Basic properties of harmonic maps into NPC spaces --- p.21 / Chapter 4.1 --- Monotonicity formula --- p.21 / Chapter 4.2 --- Approximately differentiable maps --- p.24 / Chapter 4.3 --- Local properties of harmonic maps --- p.28 / Chapter 5 --- Existence and uniqueness of harmonic maps in a ho- motopy class --- p.33 / Chapter 5.1 --- Convexity properties of the energy functional --- p.33 / Chapter 5.2 --- Existence and Uniqueness Theorem --- p.37 / Chapter 6 --- Homogeneous approximating maps --- p.40 / Chapter 6.1 --- Regular homogeneous map --- p.40 / Chapter 6.2 --- Homogeneous approximating map --- p.46 / Chapter 7 --- More results on regularity --- p.52 / Chapter 7.1 --- Intrinsically differentiable maps --- p.52 / Chapter 7.2 --- Good homogeneous approximating map --- p.62 / Chapter 8 --- Harmonic maps into building-like complexes --- p.65 / Chapter 8.1 --- F-connected complex --- p.65 / Chapter 8.2 --- Regularity and the Bochner technique --- p.66 / Bibliography --- p.75

by Tsoi, Man. / Thesis submitted in: July 1999. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 56-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminary --- p.12 / Chapter 2.1 --- Lie Group and Lie Algebra --- p.12 / Chapter 2.2 --- Harmonic Maps --- p.15 / Chapter 2.3 --- Some Factorization theorems --- p.17 / Chapter 3 --- A Survey on Unlenbeck's Results --- p.22 / Chapter 3.1 --- Preliminary --- p.24 / Chapter 3.2 --- Extended Solutions --- p.26 / Chapter 3.3 --- The Variational Formulas for the Extended Solutions --- p.30 / Chapter 3.4 --- "The Representation of A(S2, G) on holomorphic maps C* → G" --- p.33 / Chapter 3.5 --- An Action of G) on extended solutions and Backlund Transformations --- p.39 / Chapter 3.6 --- The Additional S1 Action --- p.42 / Chapter 3.7 --- Harmonic Maps into Grassmannians --- p.43 / Chapter 4 --- Harmonic Maps into Compact Lie Groups --- p.47 / Chapter 4.1 --- Symmetry group of the harmonic map equation --- p.48 / Chapter 4.2 --- A New Formulation --- p.49 / Chapter 4.3 --- "Harmonic Maps into Grassmannian, Another Point of View" --- p.53 / Bibliography

Cheng Man Chuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 78-80). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Different Notions of Negative Curvatures for Kahler Manifolds --- p.3 / Chapter 2.1 --- Definitions --- p.3 / Chapter 2.2 --- Adequate negativity of curvatures of classical domains of type I --- p.13 / Chapter 2.3 --- Adequate negativity of curvatures of classical domains of type IV --- p.23 / Chapter 2.4 --- Structure of complex semi-simple Lie algebra and its relation with Hermitian symmetric spaces --- p.28 / Chapter 2.5 --- Adequate negativity of curvatures of classical domains of type II and III --- p.34 / Chapter 2.6 --- Adequate negativity of curvatures of the two excep- tional bounded symmetric domains --- p.45 / Chapter 3 --- Complex-analyticity of Harmonic Maps between Compact Kahler Manifolds --- p.50 / Chapter 3.1 --- Existence of harmonic maps --- p.50 / Chapter 3.2 --- A Bochner type identity --- p.51 / Chapter 3.3 --- Complex-analyticity of harmonic maps --- p.58 / Chapter 3.4 --- Strong rigidity theorems --- p.62 / Chapter 3.5 --- Some further results from the Bochner technique --- p.64 / Chapter 4 --- Generalization to the Non-compact case --- p.67 / Chapter 4.1 --- A strong rigidity theorem for non-compact Kahler manifolds --- p.67 / Chapter 4.2 --- An existence theorem of harmonic map for Rieman- nian manifolds of finite volumes --- p.69 / Chapter 4.3 --- Bochner formula in the non-compact case --- p.71 / Bibliography --- p.78

by Wu Fung Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 92-95). / Acknowledgements --- p.i / Notations --- p.ii / Introduction --- p.1 / Chapter 1 --- Preliminaries --- p.8 / Chapter 1.1 --- Formulations of Harmonic Maps --- p.8 / Chapter 1.2 --- Function Spaces --- p.11 / Chapter 1.3 --- Penalized Equations --- p.13 / Chapter 2 --- Main Lemmas --- p.15 / Chapter 2.1 --- Short Time Existence --- p.16 / Chapter 2.2 --- Energy Inequalities --- p.18 / Chapter 2.3 --- The Monotonicity Inequalities --- p.23 / Chapter 2.4 --- e - Regularity Theorem --- p.30 / Chapter 3 --- The Compact Case --- p.39 / Chapter 3.1 --- Existence and Regularity for dim M = 2 --- p.39 / Chapter 3.2 --- Existence and Regularity for dim M ≥ 2 --- p.49 / Chapter 3.3 --- Blow-up Results --- p.61 / Chapter 3.4 --- Existence of Harmonic maps --- p.69 / Chapter 4 --- The Noncompact Case --- p.74 / Chapter 4.1 --- Heat-flows from Rm --- p.75 / Chapter 4.2 --- Basic Lemmas --- p.77 / Chapter 4.3 --- Nonpositive Curvature Target Manifolds --- p.83 / Chapter 4.4 --- Dirichlet Problem at Infinity --- p.88 / Bibliography --- p.92

Given a surface of genus g with fundamental group π, a representation of π into PSL(2,R) is a homomorphism that assigns to each generator of π an element of P SL(2, R). The group P SL(2, R) acts on Hom(π, P SL(2, R)) by conjugation. Define therepresentationspaceRg tobethequotientspaceHom(π,PSL(2,R))\PSL(2,R). Associated to each representation ρ is a number e(ρ) called its Euler class. Goldman showed that the space Rg has components that can be indexed by Euler classes of rep- resentations, and that there is one component for each integer e satisfying |e| ≤ 2g−2. The two maximal components correspond to Teichmu ̈ller space, the space of isotopy classes of hyperbolic structures on a surface. Teichmu ̈ller space is known to be homeomorphic to a ball of dimension 6g − 6. The other components of Rg are not as well understood. The theory of harmonic maps between non-positively curved manifolds has been used to study Teichmu ̈ller space. Given a harmonic map between hyperbolic surfaces, there is an associated quadratic differential on the domain surface called the Hopf differential. Wolf, following Sampson, proved that via the Hopf differential, harmonic maps parametrize Teichmu ̈ller space. This thesis extends his work to the case of branched hyperbolic structures, which correspond to certain elements in non- maximal components of representation space. More precisely, a branched hyperbolic structure is a pair (M, σ|dz|2) where M is a compact surface of genus g and σ|dz|2 is a hyperbolic metric with integral order cone singularities at a finite number of points expressed in terms of a conformal parameter. Fix a base surface (M, σ|dz|2). For each target surface (M, ρ|dw|2) with the same number and orders of cone points as (M,σ|dz|2), there is a unique harmonic map w : (M,σ|dz|2) → (M,ρ|dw|2) homotopic to the identity that fixes the cone points of M pointwise. Thus we may define another map from the space of branched hyperbolic structures with the same number and orders of cone points to the space of meromorphic quadratic differentials on the base surface M. This map, Φ, takes the harmonic map w associated with a metric ρ|dw|2 to the Hopf differential of w. This thesis shows that the map Φ is injective.