Constructions of harmonic maps between Hadamard manifolds

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$Tohoku Math. J. First Series General Index$

On the other hand, the (m + 1)-dimensional real hyperbolic space RH m+1 is represented as the warped product manifold ([0, ∞) × S m ,dt 2 + (sinh 2 t)g S m). Thus, for an eigenmap ϕ : S m → S n , we may consider an equivariant map u : ([0, ∞) × S m ,dt 2 + (sinh 2 t)g S m) ∋ (t, θ) ↦→ (r(t),ϕ(θ)) ∈ ([0, ∞) × S n ,dr 2 + (sinh 2 r)g S n). Then the harmonicity of u is also reduced to the following ordinary differential equation in r = r(t) defined on [0, ∞): ¨r(t)+m cosh t sinh r(t) cosh r(t) ṙ(t) − µ2 sinh t sinh 2 t where e(ϕ) =µ 2 /2 denotes the energy density function of ϕ. In Chapter 1, we shall investigate the following ordinary differential equation defined on [0, ∞), which stems from the study of such harmonic maps: { f ¨r(t)+ p ˙ 1 (t) f 1 (t) + q f ˙ } { 2 (t) ṙ(t) − µ 2 h 1(r(t))h ′ 1(r(t)) + ν 2 h } 2(r(t))h ′ 2(r(t)) =0, f 2 (t) f 1 (t) 2 f 2 (t) 2 where p, q, µ and ν are some constants. We first prove the short time existence of solutions following the method due to Baird ([3]). However, it seems to the present author that his proof needs some refinements, in particular, on the life span of solutions. In fact, under appropriate conditions for the given functions f i (t) and h i (r), we can elaborate his method and obtain a complete proof of the short time existence. Moreover, if we pose a condition on the growth rate of the function f i (t) ast goes to infinity, then such a short time solution can be extended to a global solution, which ensures the existence of an equivariant harmonic map as well as a harmonic map defined by the join. Applying these results, we can prove that there exists a family of harmonic maps from the real hyperbolic space RH m onto itself for m ≥ 3, which is parameterized by Z. =0, 7
8 Equivariant harmonic maps have been studied by many authors. For example, Kasue and Washio ([22]) investigated equivariant harmonic maps of the form u : ([0, ∞) × S m ,dt 2 + f(t) 2 g S m) ∋ (t, θ) ↦→ (r(t),ϕ(θ)) ∈ ([0, ∞) × S n ,dr 2 + h(r) 2 g S n). Here ϕ : S m → S n is an eigenmap, and f = f(t) and h = h(r) are warping functions. Since they constructed a comparison function to show the global existence of a solution for the original equation, the growth order of f and h are rather restricted. Thus one can not apply their argument to the case of real hyperbolic spaces. 3) For Hadamard manifolds (M,g) and (M ′ ,g ′ ), one can consider the Eberlein-O’Neill compactifications M and M ′ of M and M ′ , respectively, by adding their ideal boundaries, which are defined to be the spheres at infinity given by the asymptotic classes of geodesic rays. Then one can set up the Dirichlet problem at infinity for harmonic maps, which, for a given boundary map f : ∂M → ∂M ′ , consists of the existence of a harmonic map u : M → M ′ which assumes the boundary value f continuously. This problem can be regarded as a generalization of Hamilton’s work [19] on the Dirichlet problem for harmonic maps between compact Riemannian manifolds with boundary to these noncompact Riemannian manifolds. However, since the Riemannian metrics under consideration blow up at the ideal boundaries, there appear much difficulties in analyzing the boundary behavior of solutions of the harmonic map equation. The first progress to the above problem was accomplished around 1990’s by Li-Tam [25] [26] [27] and Akutagawa [1]. Recall that most typical examples of Hadamard manifolds are the rank one symmetric spaces of noncompact type, that is, the real, the complex and the quaternion hyperbolic spaces and the Cayley hyperbolic plane. In their work, Li and Tam [25] [26] [27] solved the Dirichlet problem at infinity for harmonic maps between real hyperbolic spaces. In particular, exploiting the heat equation method, they established a general theory for the existence and uniqueness of solutions to this problem. For example,
Page 1 and 2: ISSN 1343-9499 TOHOKU MATHEMATICAL
Page 3 and 4: Constructions of harmonic maps betw
Page 5 and 6: 2 Abstract In this thesis, we prese
Page 7 and 8: 4 Introduction Let (M,g) and (M ′
Page 9: 6 that for any point z ∈ S m+n−
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
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2.3. CONSTRUCTIONS OF EQUIVARIANT H
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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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