Constructions of harmonic maps between Hadamard manifolds

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$Tohoku Math. J. First Series General Index$

3.1. DAMEK-RICCI SPACES 61 where [ , ] Ò is the Lie bracket on n defined by (3.1.1). In this way, s becomes a Lie algebra with an inner product. The simply connected Lie group S associated with s, equipped with the left invariant metric g S , is called a Damek-Ricci space. Example. Let v = R 2k , u = R 2 , and µ(z,v) =(−y, x),µ(w, v) =(x, y), where {z,w} is an orthonormal basis of R 2 and v =(x, y) ∈ R k ⊕ R k . Then n is a (classical) Heisenberg algebra and (S, g S ) is isometric to the complex hyperbolic space whose sectional curvature K satisfies −4 ≤ K ≤−1. The quaternion hyperbolic space and the Cayley hyperbolic plane are also Damek-Ricci spaces. These three spaces are called classical (cf. [8]). Thus Damek-Ricci spaces are generalizations of rank one symmetric spaces of noncompact type. Let ∇ be the Levi-Civita connection on (S, g S ). Using the formula 2〈∇ X Y,Z〉 = 〈[X, Y ],Z〉 + 〈[Z, X],Y〉 + 〈[Z, Y ],X〉 for X, Y, Z ∈ s, one obtains ⎧ ∇ h X =0, ∇ X h = −[h, X], ⎪⎨ ∇ v v ′ = 〈v, v ′ 〉h + 1 2 [v, v′ ], ∇ v z = ∇ z v = −µ(z,v), ⎪⎩ ∇ z z ′ =2〈z,z ′ 〉h, where X ∈ s,v,v ′ ∈ v,z,z ′ ∈ z and µ is the composition of quadratic forms in the definition of n. Then the following results have been proved. Theorem 3.1.2. ([7], [8]). (1) A Damek-Ricci space is a symmetric space if and only if it is classical (2) A Damek-Ricci space has strictly negative sectional curvature if and only if it is classical ([13]). In fact, Damek constructed a simple example of Damek-Ricci space whose sectional curvature attains zero for some two-dimensional plane. Other examples have been given
62 CHAPTER 3. DAMEK-RICCI SPACES in [5]. The second assertion was first proved by Boggino [4]. However, his proof needed a refinement which had been accomplished by Dotti([13]), see also [23]. The above theorem claims that every non-symmetric Damek-Ricci space has two-dimensional planes for which the sectional curvature vanishes. We refer to [5] for further information about the geometry, in particular about the sectional curvature, of the Damek-Ricci spaces. Finally, we estimate the bottom spectrum λ 1 (S) of the Laplace-Beltrami operator for S with the left invariant metric. Lemma 3.1.3 ([9]). Let (r, ω) be the polar coordinate on (S, g S ) around an origin. Then the volume form dv S of S is given by dv S =2 −m (sinh r) n (sinh 2r) m drdσ(ω), where n = dim v,m = dim z and dσ(ω) is the surface element of the standard unit sphere S n+m . From this lemma and the fact λ 1 (S) ≥ inf(∆ S r) 2 /4, where ∆ S Laplace-Beltrami operator, we obtain the following denotes the positive Corollary 3.1.4. λ 1 (S) ≥ 1 4 (n +2m)2 . Note. Damek and Ricci [10] proved that the equality holds in the above inequality. 3.2 Harmonic maps between Damek-Ricci spaces In this section, following Donnelly [12], we shall prove the existence and uniqueness of solutions of the Dirichlet problem at infinity for harmonic maps.
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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−
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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 63: 60 CHAPTER 3. DAMEK-RICCI SPACES a
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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Constructions of harmonic maps between Hadamard manifolds

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