Constructions of harmonic maps between Hadamard manifolds

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$Tohoku Math. J. First Series General Index$

CHAPTER 3 Dirichlet problem at infinity for harmonic maps between Damek-Ricci spaces 3.1 Damek-Ricci spaces 3.1.1 Generalized Heisenberg algebra We start this section with a brief review of generalized Heisenberg algebras due to Kaplan [21]. Let (n, 〈 , 〉 Ò ) be a 2-step nilpotent Lie algebra with inner product 〈 , 〉 Ò , z its center and v the orthogonal complement of z in n with respect to 〈 , 〉 Ò . Since n is 2-step nilpotent, ad v| Ú is a map from v to z for any v ∈ v. We set k v := {u ∈ v | ad v(u) =[v, u] =0} and consider the orthogonal decomposition v = k v ⊕ v v with respect to 〈 , 〉 Ò . We call (n, 〈 , 〉 Ò )a generalized Heisenberg algebra if ad v| Úv : v v → z is a surjective isometry for any unit vector v ∈ v. Generalized Heisenberg algebras can be constructed systematically in the following fashion: Let (u, 〈 , 〉 Ù ) and (v, 〈 , 〉 Ú ) be real vector spaces equipped with inner product. Let µ : u × v → v be a composition of quadratic forms, that is, µ is a bilinear map satisfying |µ(u, v)| Ú = |u| Ù |v| Ú for all u ∈ u and v ∈ v. Note that µ : u × v → v satisfies µ(u 0 ,v)=v for some u 0 ∈ u. Indeed, for any given u 0 ∈ u, set T : v → v by T (v) :=µ(u 0 ,v). Then µ ′ (u, v) :=µ(u, T −1 (v)) satisfies µ ′ (u 0 ,v)=v. Hence we may suppose µ(u 0 ,v)=v. Define φ : v × v → u by 〈u, φ(v, v ′ )〉 Ù = 〈µ(u, v),v ′ 〉 Ú . Let z be the orthogonal complement of Ru 0 = {ru 0 | r ∈ R} in u, π: u → z the orthogonal projection and n the direct sum n := v ⊕ z of v and z with the natural inner product. Define 59
60 CHAPTER 3. DAMEK-RICCI SPACES a Lie bracket on n by (3.1.1) [v + z,v ′ + z ′ ]:=π ◦ φ(v, v ′ ). Theorem 3.1.1 ([21]). n, equipped with the above inner product and a Lie bracket, is a generalized Heisenberg algebra. Conversely, any generalized Heisenberg algebra arises in this manner. Regarding the existence of a composition of quadratic forms, we remark the following. Let ρ be a function defined on positive integers by ρ(n) :=8p +2 q if n = (odd number) × 2 4p+q , 0 ≤ q ≤ 3. Then it is known by Hurwitz, Radon and Eckmann that a composition of quadratic forms µ : u × v → v exists if and only if 0 < dim u ≤ ρ(dim v). Therefore we may conclude (see [21]): Fact. Given integers m, n > 0, there exists an (n + m)-dimensional generalized Heisenberg algebra with m-dimensional center if and only if 0
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ISSN 1343-9499 TOHOKU MATHEMATICAL
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Constructions of harmonic maps betw
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2 Abstract In this thesis, we prese
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4 Introduction Let (M,g) and (M ′
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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
Page 54 and 55: 2.2. REDUCTION OF HARMONIC MAP EQUA
Page 56 and 57: 2.3. CONSTRUCTIONS OF EQUIVARIANT H
Page 58 and 59: 2.3. CONSTRUCTIONS OF EQUIVARIANT H
Page 60: 2.3. CONSTRUCTIONS OF EQUIVARIANT H
Page 65 and 66: 62 CHAPTER 3. DAMEK-RICCI SPACES in
Page 67 and 68: 64 CHAPTER 3. DAMEK-RICCI SPACES wh
Page 69 and 70: 66 CHAPTER 3. DAMEK-RICCI SPACES (2
Page 71 and 72: 68 CHAPTER 3. DAMEK-RICCI SPACES Si
Page 73 and 74: 70 CHAPTER 3. DAMEK-RICCI SPACES Th
Page 76 and 77: CHAPTER 4 Non-existence of proper h
Page 78 and 79: 4.1. PROOF OF THEOREM 75 Moreover,
Page 80 and 81: 4.1. PROOF OF THEOREM 77 Propositio
Page 82 and 83: 4.2. A COUNTER EXAMPLE 79 where (y,
Page 84 and 85: 4.2. A COUNTER EXAMPLE 81 Hence we
Page 86 and 87: Bibliography [1] K. Akutagawa, Harm
Page 88 and 89: BIBLIOGRAPHY 85 [21] A. Kaplan, Fun
Page 90 and 91: BIBLIOGRAPHY 87 [44] R. Wood, Polyn
Page 92: No.16 Tatsuo Konno: On the infinite
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