By Lin F., Wang C.
This ebook presents a wide but accomplished creation to the research of harmonic maps and their warmth flows. the 1st a part of the ebook comprises many vital theorems at the regularity of minimizing harmonic maps by way of Schoen-Uhlenbeck, desk bound harmonic maps among Riemannian manifolds in greater dimensions by way of Evans and Bethuel, and weakly harmonic maps from Riemannian surfaces via Helein, in addition to at the constitution of a novel set of minimizing harmonic maps and desk bound harmonic maps by way of Simon and Lin.The moment a part of the publication incorporates a systematic insurance of warmth stream of harmonic maps that incorporates Eells-Sampson's theorem on worldwide gentle options, Struwe's nearly average recommendations in measurement , Sacks-Uhlenbeck's blow-up research in measurement , Chen-Struwe's lifestyles theorem on in part gentle suggestions, and blow-up research in greater 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 drawn to geometric partial differential equations and geometric research.
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38). ✷ 30 CHAPTER 2. 5 holds with = i −1 . Then we have Sj (u) = ∪i≥1 Sj,i . Next for any integer q ≥ 1 we let Sj,i,q = x ∈ Sj,i : Θn−2 (u, x) ∈ ( q−1 q , ] i i so that Sj (u) = ∪i,q Sj,i,q . 41) for some j-dimensional subspace Ly,ρ of Rn . 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) .
45) L2 (S n−1 ) < δ, where τS n−1 (ψ) = ∆S n−1 ψ + A(ψ) (∇S n−1 ψ, ∇S n−1 ψ) is the tension field of ψ from S n−1 to (N, h). 1. For simplicity, assume M = Ω ⊂ R n and x0 = 0 ∈ Ω. First recall, by the definition of minimizing tangent map, that there is a seqeuence ρi ↓ 0 such that the rescaled maps uρi (≡ u(ρi x)) converge in x . By the assumption, we have φ0 ∈ C ∞ (S n−1 , N ) is a smooth H 1 to φ(x) = φ0 |x| harmonic map. Thus for any η > 0 there is a sufficiently large j such that for ρ = ρ j we have |uρ − φ|2 < η 2 .
5. 50) |∇˜ u|2 satisfies Θ(˜ u, 0) = Θ(u, x0 ) = Θ(φ, 0) = 1 n−2 S n−1 |∇S n−1 φ|2 . 20) we actually showed that u ˜ satisfies (n − 2) B1 |∇˜ u|2 = |∇˜ u|2 − 2| S n−1 ∂u ˜ 2 | ∂r |∇S n−1 u ˜ |2 . 52) |∇S n−1 u ˜|2 − |∇S n−1 φ|2 . 8) for u ˜ in terms of spherical coordinates x r = |x|, ω = |x| gives r 1−n ∂ ∂r r n−1 ∂u ˜ ∂r + r −2 τS n−1 (˜ u) + A(˜ u) ∂u ˜ ∂u ˜ , ∂r ∂r = 0. 54) < η. 4 To handle the second order derivative term, we observe that the domain deformation d |t=0 u ˜ ((1 + t)x) = r ∂∂ru˜ satisfies the u ˜ ((1 + t)x) is again a harmonic map.