Constructions of harmonic maps between Hadamard manifolds

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$Tohoku Math. J. First Series General Index$

3.2. HARMONIC MAPS BETWEEN DAMEK-RICCI SPACES 69 Then there exists an h ∈ C 2,ε ( ¯S, ¯S ′ ) such that h = f at the ideal boundary and ‖τ(h)‖ = O(y ε ), where ‖τ(h)‖ is the norm of the tension field in the Riemannian metric. Proof. Let φ>0 be a unique solution of (n +2m)φ 4 − n∑ ∑n ′ (f β j )2 φ 2 − 2 j=1 β=1 n+m ∑ n ∑ ′ +m ′ j=n+1 β=n ′ +1 (f β j )2 =0. Since f ∈ C 3,ε , we have φ ∈ C 2,ε (∂S). Set h(y, n) :=(yφ(n),f(n)). Then h ∈ C 2,ε (∂S,∂S ′ ). Moreover, it is easy to verify that h satisfies ⎧ h 0 0 = φ, ⎪⎨ h α 0 =0 (1≤ α ≤ n ′ + m ′ ), h β 00 =0 (n ′ +1≤ β ≤ n ′ + m ′ ), ⎪⎩ h γ j0 =0 (1≤ j ≤ n, n′ +1≤ γ ≤ n ′ + m ′ ), and h = f at the ideal boundary. Lemma 3.2.6 and the explicit expression for the metric then imply ‖τ(h)‖ = O(y ε ). Next we construct a comparison function. Lemma 3.2.8. For sufficiently large r 0 and some constant s, define ⎧ ⎪⎨ e −sr 0 (r ≤ r 0 ), ψ(r) := ⎪⎩ e −sr (r ≥ r 0 ). If 0
70 CHAPTER 3. DAMEK-RICCI SPACES Theorem 3.2.9 (Existence Theorem). Suppose that f ∈ C 3,ε (∂S,∂S ′ )(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−
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8 Equivariant harmonic maps have be
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10 vanishing sectional curvatures f
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CHAPTER 1 Ordinary differential equ
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1.1. LOCAL EXISTENCE OF SOLUTIONS T
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1.1. LOCAL EXISTENCE OF SOLUTIONS T
Page 22 and 23: 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 and 64: 60 CHAPTER 3. DAMEK-RICCI SPACES a
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: 68 CHAPTER 3. DAMEK-RICCI SPACES Si
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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