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By Lin F., Wang C.

This publication offers a vast but finished advent to the research of harmonic maps and their warmth flows. the 1st a part of the booklet includes many vital theorems at the regularity of minimizing harmonic maps through Schoen-Uhlenbeck, desk bound harmonic maps among Riemannian manifolds in greater dimensions through Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces via Helein, in addition to at the constitution of a unique set of minimizing harmonic maps and desk bound harmonic maps by means of Simon and Lin.The moment a part of the booklet incorporates a systematic insurance of warmth move of harmonic maps that comes with Eells-Sampson's theorem on worldwide tender recommendations, Struwe's nearly typical suggestions in size , Sacks-Uhlenbeck's blow-up research in size , Chen-Struwe's life theorem on partly tender ideas, and blow-up research in larger dimensions by way of Lin and Wang. The e-book can be utilized as a textbook for the subject process complex graduate scholars and for researchers who're drawn to geometric partial differential equations and geometric research.

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Thus the conclusions of this theorem follow easily. 3: Note, since φ ∈ C ∞ (Rn \ {0}, N ) is a tangent map of u at 0, that for any > 0 there is ρ0 > 0 such that u(ρ0 ·) − φ(·) C 2 (B1 \B 1 ) ≤ . 2 Hence, by considering uρ0 (x) = u(ρ0 x), we may assume that there exists 0 < λ = λ( , n) < 1 such that u − φ 1,λ3 ≤ . 6. 6. ✷ It seems difficult to check the integrability criterion in higher dimensions. However, Gulliver-White [64] verified the integrability condition for harmonic maps from S 2 to any two dimensional Riemannian manifold N .

Any such a φ is called a minimizing tangent map of u at x . 20), φ(x) = φ 0 x |x| is homogeneous of degree zero and φ0 : S n−1 → N is a weakly harmonic map. A very important question is whether a minimizing tangent map is unique, that is, depending on the original map u and the point x0 , but independent of the rescaling sequence r i → 0. If x0 is an isolated singularity for u, then any minimizing tangent map φ of u at x0 has 0 as its only singularity. Such uniqueness would imply that the difference u(x) − φ(x − x0 ) is continuous and vanishes at x0 .

83) so that for j ≥ 1, aj (r) + λj n−1 aj (r) − 2 aj (r) = 0, 0 < r < 1. 6. INTEGRABILITY OF JACOBI FIELDS AND ITS APPLICATIONS Solving this ODE, we get aj (r) = r γj with 2−n ± 2 γj = (n − 2)2 + λj . 84) where aj , bj , cj , dj , ej ∈ R and J1 = j : λj > − (n − 2)2 4 , J2 = j : λj < − and J3 = j : λj = − (n − 2)2 4 (n − 2)2 4 and βj = Imγj , . 82), we conclude J 2 = ∅ and bj = 0 (the coefficient of ln |x|) for j ∈ J3 so that x |x| cj φj w(x) = j∈J1 It is easy to see that ψ(ω) = Moreover, note that γj ≥ γ 1 = |x|γj + j∈J3 2−n + 2 x |x| aj φj j∈J3 , x ∈ B1 \ {0}.

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