By Lin F., Wang C.

This e-book presents a large but accomplished creation 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 by means of Schoen-Uhlenbeck, desk bound harmonic maps among Riemannian manifolds in better dimensions by way of Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces by way of 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 publication features a systematic assurance of warmth circulate of harmonic maps that comes with Eells-Sampson's theorem on worldwide soft ideas, Struwe's nearly average strategies in measurement , Sacks-Uhlenbeck's blow-up research in size , Chen-Struwe's lifestyles theorem on in part gentle ideas, and blow-up research in better 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 attracted to geometric partial differential equations and geometric research.

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MINIMIZING HARMONIC MAPS IN HIGHER DIMENSIONS 25 This completes the proof. 9, we have the following compactness result of minimizing harmonic maps, due to [171] and [140]. 13 Let ui ∈ W 1,2 (Ω, N ) be a sequence of minimizing harmonic maps. 1,2 If ui → u weakly in W 1,2 (Ω, N ), then ui → u strongly in Wloc (Ω, N ) and u is a minimizing harmonic map. Proof. For any unit ball B1 ⊂⊂ Ω and a small λ ∈ (0, 1), let w ∈ H 1 (B, N ) be such that w = u on B1 \ B1−λ . By Fatou’s lemma and Fubini’s theorem, there is ρ ∈ (1 − λ0 , 1) such that lim i→∞ ∂Bρ |ui − u|2 dH n−1 = 0, ∂Bρ |∇ui |2 + |∇w|2 dH n−1 ≤ C < +∞.

3 (i) The real analyticity of N plays a crucial role for the uniqueness of minimizing tangent maps of a minimizing harmonic map at its isolated singular points. White [209] constructed a Riemannian manifold N which is not real analytic such that there exists an energy minimizing map into N that has infinitely many tangent maps at its isolated singularity. (ii) Gulliver-White [64] constructed a real analytic Riemannian manifold N and an energy minimizing map u from M to N with an isolated singular point x 0 ∈ M with the property that the convergence of u to its tangent map at x 0 has at most logarithmic decay.

Now letting P = Π Π◦Φ on Nδ on RL \ Nδ , we find that BR |∇P |2 ≤ [cLip(Π)]2 dist−2 (y, X). BR To see that the latter integral is finite, we may assume that X is an affine space of dimension at most (L − 3), so that dist−2 (y, X) BR is finite by Fubini’s theorem. g. h is a harmonic extension). However, the image of h may not be contained in N . To correct this, we compose h with a suitable projection onto N . 21, and let B ⊂ RL be a large ball containing N ∪ X. For τ < min{δ, dist(N ∪ X, ∂B)} and a point a ∈ B τ = {y ∈ RL | |y| < τ }, define Ba = B + {a} and Xa = X + {a} and the projection Pa : Ba \ Xa → N by Pa (y) = P (y − a).