By Fanghua Lin

ISBN-10: 9812779523

ISBN-13: 9789812779526

This booklet presents a large but entire creation to the research of harmonic maps and their warmth flows. the 1st a part of the ebook comprises many very important theorems at the regularity of minimizing harmonic maps by way of Schoen-Uhlenbeck, desk bound harmonic maps among Riemannian manifolds in better dimensions by means of Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces by way of Helein, in addition to at the constitution of a novel set of minimizing harmonic maps and desk bound harmonic maps by means of Simon and Lin.The moment a part of the publication incorporates a systematic assurance of warmth circulate of harmonic maps that comes with Eells-Sampson's theorem on international soft suggestions, Struwe's virtually normal ideas in size , Sacks-Uhlenbeck's blow-up research in size , Chen-Struwe's lifestyles theorem on partly delicate ideas, and blow-up research in greater dimensions by means of Lin and Wang. The booklet 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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2 This contradicts the choices of ui . ✷ Now we are ready to prove the following decay lemma (cf. 4). 76) 1,λ2 for some harmonic map φ2 ∈ C 2 (S n−1 , N ). Proof. We prove it by contradiction. Suppose that the lemma were false for some λ ≤ λ0 . Then there exist i ↓ 0, harmonic maps {φi } ⊂ C 2 (S n−1 , N ) with φi − φ C 2 (S n−1 ) ≤ 1i , and {ui } ⊂ Q(φ, i , λ), but inf ui − φ˜ λ,λ3 ˜ = 0 > 1 ui − φi : φ˜ ∈ C 2 (S n−1 , N ), τ (φ) 2 1,λ2 . 4, there is a sequence of R i ↓ 0 so that lim ui − φi i→0 x |x| = 0.

Denote 2 = E(u, B1 ). Let φ ∈ C ∞ (Rn , R+ ) be a radial mollifying function so that supp φ ⊆ B 1 and Rn φ = 1. ¯ = 21 , τ = 41 , and θ ∈ (τ, 1 ). Define h = h(r), r = |x|, to be a nondecreasing Let h 4 smooth function satisfying h(r) = ¯h for r ≤ θ, h(θ + τ ) = 0, and |h (r)| ≤ 2 1 4 . Then set u(h(x)) (x) = B1 φ(h(x)) (x − y)u(y) dy, x ∈ B1 , 2 x where φ(h(x)) = h(x)−n φ h(x) . 20), we have dist2 u(h(x)) , N ≤ 1 |Bh(x) (x)| ≤ Ch(x)2−n Hence for 0 2 Bh(x) (x) u(y) − u(h(x)) Bh(x) (x) |∇u|2 dy ≤ C 2 .

3. 32). 33) ˜ ⊂⊂ Ω, we can apply simple covering whenever B2R ⊂ Ω. 31). (3) Note that for j ≥ 1, uj (x1 , · · · , xn ) = (cos jx1 , sin jx1 ) : B ⊂ Rn → S 1 ⊂ R2 are minimizing harmonic maps that have unbounded energy on each subdomain. 3 Federer’s dimension reduction principle In this section we will first present the dimension reduction principle, which was first developed by Federer [54] in the study of area minimizing currents, of minimizing harmonic maps by [171]. 4. Here we follow the presentation by Simon [187, 188].

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