By Lin F., Wang C.
This publication presents a wide but entire creation to the research of harmonic maps and their warmth flows. the 1st a part of the ebook includes many very important theorems at the regularity of minimizing harmonic maps by way of Schoen-Uhlenbeck, desk bound harmonic maps among Riemannian manifolds in greater dimensions by means of 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 includes a systematic insurance of warmth stream of harmonic maps that comes with Eells-Sampson's theorem on worldwide gentle options, Struwe's virtually ordinary strategies in size , Sacks-Uhlenbeck's blow-up research in size , Chen-Struwe's lifestyles theorem on partly soft ideas, and blow-up research in better dimensions by means of Lin and Wang. The ebook 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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Extra resources for Analysis of harmonic maps and their heat flows
9 For any bounded domain Ω ⊂ R 3 , let u : Ω → S 2 be a minimizing harmonic map which is singular at x0 ∈ Ω. Then there exist an orthogonal rotation θ of R3 , C > 0, α ∈ (0, 1) and r0 > 0 such that u(x0 + x) − θ x |x| ≤ C |x|α for all |x| ≤ r0 . 90) To conclude this section, we would like to mention the very important work by Simon  on the rectifiability of singular set of minimizing harmonic maps in dimensions at least four. In dimensions ≥ 4, higher dimensional singularities may occur for minimizing harmonic maps, and it is a great challenge to study their structure and asymptotics.
We now recall that if L is a j-dimensional subspace of R n and for each δ ∈ (0, 18 ) we can find β = β(δ) with limδ→0 β(δ) = 0 and σ = σ(δ) ∈ (0, 1) such that for each R > 0 a 2δR-neighborhood of L ∩ BR (0) can be covered by balls BδR (yk ) with centers yk ∈ L ∩ BR (0), k = 1, · · · , Q such that Q(δR)j+β(δ) < 21 Rj+β(δ) . Now the lemma follows by using successively finer covers of A by balls. For simplicity assume A is bounded, we first take an initial cover of A by balls B ρ0 (yk ) 2 with A ∩ B ρ0 (yk ) = ∅, k = 1, · · · , Q, and let T0 = 21 ( ρ20 )j+β(δ) .
48 CHAPTER 2. REGULARITY OF MINIMIZING HARMONIC MAPS There are many new ideas in the proof of the above theorem in , and it is beyond the scope of this book to outline them. Here we just briefly mention some key steps. First, there is a new technical criteria for (n − 3)-rectifiability of a set involving an alternative at each scale between being located near some (n − 3)-plane and having fixed size gaps. Second one reduces to consider singular points a at which a tangent map φ depends on three vairable and sing(φ) = R n−3 .