By Lin F., Wang C.

ISBN-10: 9812779523

ISBN-13: 9789812779526

This e-book offers a huge but entire creation to the research of harmonic maps and their warmth flows. the 1st a part of the e-book comprises many vital theorems at the regularity of minimizing harmonic maps by way of Schoen-Uhlenbeck, desk bound harmonic maps among Riemannian manifolds in larger dimensions via Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces by means of Helein, in addition to at the constitution of a novel set of minimizing harmonic maps and desk bound harmonic maps via Simon and Lin.The moment a part of the e-book features a systematic assurance of warmth stream of harmonic maps that incorporates Eells-Sampson's theorem on international tender ideas, Struwe's nearly standard options in size , Sacks-Uhlenbeck's blow-up research in measurement , Chen-Struwe's life theorem on partly soft suggestions, and blow-up research in better dimensions via Lin and Wang. The ebook can be utilized as a textbook for the subject process complicated graduate scholars and for researchers who're attracted to geometric partial differential equations and geometric research.

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We will consider the energy of a comparison map obtained by homogeneous of degree zero extension from a point (0, · · · , 0, α), where 0 < α < 1. We use spherical polar coordinates to represent x ∈ ∂B ∩ {xn ≥ 0} by (ω, φ) ∈ S n−2 × [0, π2 ]. Let θ be the angle of S n−1 with (0, · · · , 0, α). Then we have θ = φ + sin−1 (α sin θ) . As the angle φ varies from 0 to π 2, the angle θ varies from 0 to Θ(α) = π − sin−1 1 + α2 − 12 . 4. BOUNDARY REGULARITY FOR MINIMIZING HARMONIC MAPS The distance between x and (0, · · · , 0, α) is R(φ, α) = (α − cos φ)2 + sin2 α comparison map is given by 1 2 33 .

7 (see also [83]). 44) By iterating this lemma, we then get the following consequence. 5 Suppose that Ω ∈ C 1 , b ∈ ∂Ω, φ : ∂Ω → N is C 1 near b. If R > 0 is small enough and u ∈ H 1 (Ω, N ) is a minimizing harmonic map with u| ∂Ω = φ and R2−n B2R (b)∩Ω |∇u|2 ≤ 2 then u ∈ C α (Ω ∩ BR (b), N ) for some α ∈ (0, 1). The final step for the full boundary regularity of minimizing harmonic maps is to rule out any possible boundary tangent map. For any a ∈ ∂Ω and r i ↓ 0, define ua,ri (x) = u(a + ri x), x ∈ ri−1 (Ω \ {a}) → N.

Any smooth map v : S 2 → N which does not extend continuously to B 3 is homotopic to a sum of smooth harmonic maps uj : S 2 → N . Proof. Since v is smooth, it is easy to see that v¯(x) = v x |x| : B 3 → N is an extension of v of finite energy. Let u ∈ H 1 (B 3 , N ) be a minimizing harmonic map with u|∂B 3 = v. 1 we have that there are finite points x1 , · · · , xp (p ≥ 1) in B 3 such that u ∈ C ∞ B \ {x1 , · · · , xp }, N . Moreover each xj , 1 ≤ j ≤ p is associated with a minimizing tangent map φ j x |x| : R3 → N , which is smooth when restricted to S 2 .

### Analysis of harmonic maps and their heat flows by Lin F., Wang C.

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