Constructions of harmonic maps between Hadamard manifolds

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$Tohoku Math. J. First Series General Index$

1 Acknowledgments The author wishes to express his sincere gratitude to Professor Seiki Nishikawa for his valuable discussions and encouragement throughout the preparation of this thesis. He also would like to express his deep appreciation to Professor Hajime Urakawa who directed his interest to the study of harmonic maps, Professors Izumi Takagi and Takeyuki Nagasawa for their invaluable advice on the theory of ordinary differential equations, and Professor Kazuo Akutagawa who introduced him to some relevant topics in the field.
2 Abstract In this thesis, we present several effective methods of constructing harmonic maps between Hadamard manifolds. In particular, in the case of real hyperbolic spaces, we reduce the harmonic map equation to an ordinary differential equation defined on the nonnegative real line, and by showing the existence of solutions for a boundary value problem of this equation, we construct a variety of new harmonic maps between real hyperbolic spaces. Furthermore, following the idea due to Donnelly, we establish the existence and uniqueness result for proper harmonic maps between Damek-Ricci spaces, which are a generalization of rank one symmetric spaces of noncompact type. A non-existence result for proper harmonic maps from complex hyperbolic spaces to real hyperbolic spaces is also proved.
Page 1 and 2: ISSN 1343-9499 TOHOKU MATHEMATICAL
Page 3: Constructions of harmonic maps betw
Page 7 and 8: 4 Introduction Let (M,g) and (M ′
Page 9 and 10: 6 that for any point z ∈ S m+n−
Page 11 and 12: 8 Equivariant harmonic maps have be
Page 13 and 14: 10 vanishing sectional curvatures f
Page 16 and 17: CHAPTER 1 Ordinary differential equ
Page 18 and 19: 1.1. LOCAL EXISTENCE OF SOLUTIONS T
Page 34 and 35: 1.2. GLOBAL PROPERTIES OF SOLUTIONS
Page 44 and 45: 1.3. ADDENDUM 41 Theorem 1.2.11. Th
Page 46 and 47: CHAPTER 2 Equivariant harmonic maps
Page 48 and 49: 2.1. EIGENMAPS 45 Following a metho
Page 50 and 51: 2.1. EIGENMAPS 47 Then the restrict
Page 52 and 53: 2.1. EIGENMAPS 49 exists a full ort
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2.2. REDUCTION OF HARMONIC MAP EQUA
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2.3. CONSTRUCTIONS OF EQUIVARIANT H
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Page 60:
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60 CHAPTER 3. DAMEK-RICCI SPACES a
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62 CHAPTER 3. DAMEK-RICCI SPACES in
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64 CHAPTER 3. DAMEK-RICCI SPACES wh
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66 CHAPTER 3. DAMEK-RICCI SPACES (2
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68 CHAPTER 3. DAMEK-RICCI SPACES Si
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70 CHAPTER 3. DAMEK-RICCI SPACES Th
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CHAPTER 4 Non-existence of proper h
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4.1. PROOF OF THEOREM 75 Moreover,
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4.1. PROOF OF THEOREM 77 Propositio
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4.2. A COUNTER EXAMPLE 79 where (y,
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4.2. A COUNTER EXAMPLE 81 Hence we
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Bibliography [1] K. Akutagawa, Harm
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BIBLIOGRAPHY 85 [21] A. Kaplan, Fun
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BIBLIOGRAPHY 87 [44] R. Wood, Polyn
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No.16 Tatsuo Konno: On the infinite
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Constructions of harmonic maps between Hadamard manifolds

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