By Lin F., Wang C.

ISBN-10: 9812779523

ISBN-13: 9789812779526

This ebook presents a large but finished advent to the research of harmonic maps and their warmth flows. the 1st a part of the e-book comprises many very important 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 through Helein, in addition to at the constitution of a novel set of minimizing harmonic maps and desk bound harmonic maps through Simon and Lin.The moment a part of the publication incorporates a systematic assurance of warmth move of harmonic maps that incorporates Eells-Sampson's theorem on worldwide gentle ideas, Struwe's virtually ordinary suggestions in size , Sacks-Uhlenbeck's blow-up research in size , Chen-Struwe's lifestyles theorem on partly gentle options, and blow-up research in better dimensions through Lin and Wang. The publication 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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43) ✷ Next we show that there is no minimizing tangent maps at the boundary. 3 Any minimizing harmonic map u 0 ∈ H 1 (B + , N ) that is homogeneous of degree zero and that is constant on B ∩ {x n = 0} must be constant. Proof. We follow the construction by Hardt-Lin [83]. 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 ].

3 For j = 0, · · · , n − 3, dim H (Sj (u)) ≤ j. 4. 4 For n ≥ 3, let u ∈ H 1 (Ω, N ) be a minimizing harmonic map. Then sing(u) has Hausdorff dimension at most (n − 3), and is discrete for n = 3. 3. FEDERER’S DIMENSION REDUCTION PRINCIPLE Proof. 3. For the second conclusion, we argue by contradiction. Suppose it were false. Then there exist {x i } and x0 in sing(u) such that xi → x0 . Set ri = |xi − x0 | → 0. Define vi (x) = u(x0 + ri x) : B2 → N . Then we have lim i→∞ B2 |∇vi |2 = lim i→∞ ri2−n B2ri (x0 ) |∇u|2 = 2n−2 Θn−2 (u, x0 ).

This chapter is organized as follows. 1, we present the classical theorem on weakly harmonic maps into regular balls by Hildebrandt-Kaul-Widman [95]. 2, we present the regularity of weakly harmonic maps by H´elein [90, 91, 92]. In [93], H´elein has given a detailed account of his important works. 3, we present the partial regularity of stationary harmonic maps. 4, we present some optimal partial regularity for stable-stationary harmonic maps. 1 Weakly harmonic maps into regular balls In this section, we will present the fundamental theorem, due to Hildebrandt-KaulWidman [95], on the regularity of weakly harmonic maps whose images are contained in any regular ball of N .