Constructions of harmonic maps between Hadamard manifolds

More documents

Recommendations

Info

$Tohoku Math. J. First Series General Index$

1.2. GLOBAL PROPERTIES OF SOLUTIONS 35 for any t 0 >T 0 . Then, by Theorem 1.2.3, there exists globally a solution r = r(t) to the equation (1.1.1) with the boundary condition (1.1.2) satisfying r(t 0 )=l − ε>0. Let r(t; t 0 ) denote this solution. Since it is bounded and increasing, the limit r(∞; t 0 ) exists and 0 ···→l, lim ¯r j (t) =¯l j . t→∞
36 CHAPTER 1. ORDINARY DIFFERENTIAL EQUATIONS It follows from Corollary 1.1.3 that r 1 (t) 0, and define two positive numbers α and ᾱ by α = lim i→∞ r i (t 0 ) and ᾱ = lim j→∞ ¯r j (t 0 ). Choose α such that α ≤ α ≤ ᾱ, and let r α be a unique solution to the equation (1.1.1) with the boundary condition (1.1.2) satisfying r α (t 0 )=α. Making use <strong>of</strong> Corollary 1.1.3 again, we get r i (t) 0 there exists a solution to the equation (1.1.1) with the boundary condition (1.1.2) satisfying r(t) → +∞ as t → T. We suppose that h(s) = max{h 1 (s),h 2 (s)} satisfies (1.2.5) ∫ ∞ ds h(s) < ∞. Note that if the assumption (1.2.5) is false, then any solution is bounded (Corollary 1.2.4). For t 0 > 0 and r 0 > 0, let r = r(t) be a solution to the equation (1.1.1) with the boundary condition (1.1.2) satisfying r(t 0 )=r 0 . Such a solution exists uniquely. We denote ṙ(t 0 )by β(r 0 ,t 0 ), which is uniquely determined by r 0 and t 0 . We define φ(r 0 ,t 0 )= where ∫ ∞ r 0 [γ 1 (t 0 ) 2 {h 1 (s) 2 − h 1 (r 0 ) 2 } + γ 2 (t 0 ) 2 {h 2 (s) 2 − h 2 (r 0 ) 2 } + β(r 0 ,t 0 ) 2 ] −1/2 ds, γ 1 (t 0 )= µ f 1 (t 0 ) and γ 2 (t 0 )= ν f 2 (t 0 ).
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 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
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 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
show all

Constructions of harmonic maps between Hadamard manifolds

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?