Download e-book for iPad: Analysis of harmonic maps and their heat flows by Lin F., Wang C.

By Lin F., Wang C.

This booklet offers a extensive but finished 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 larger 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 through Simon and Lin.The moment a part of the ebook features a systematic assurance of warmth stream of harmonic maps that comes with Eells-Sampson's theorem on international tender options, Struwe's nearly average recommendations in measurement , Sacks-Uhlenbeck's blow-up research in measurement , Chen-Struwe's life theorem on in part soft strategies, and blow-up research in better dimensions via Lin and Wang. The booklet can be utilized as a textbook for the subject process complex graduate scholars and for researchers who're drawn to geometric partial differential equations and geometric research.

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By Alexander duality and Eilenberg extension theorem we can extend the identity map of N δ to W ∪ Nδ and hence RL \ X. Denote such an extension map as Φ. Since the c constructions are piecewise linear, we can assume |∇Φ|(y) ≤ dist(y,X) for a constant depending only on N . 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.

13 1 (Rn , N ). e. 5. UNIQUENESS OF MINIMIZING TANGENT MAPS then any tangent map Φ of u at a is a minimizing harmonic map on R n+ , which is homogeneous of degree zero and constant on ∂R n+ . 3. 1 is proven. ✷ Now we present an application (see [172]) of the regularity theorems we have developed for minimizing harmonic maps. 6 Suppose that N is compact without boundary. 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 .

Therefore for R0 > 0 sufficiently small, v lies in the δ-neighborhood of N , where ΠN : Nδ → N is the smooth nearest point projection, and u ˆ = Π N ◦v is a comparison map for u. Hence by the minimality we have + BR (0) |∇u|2 ≤ ≤ + BR (0) |∇ˆ u|2 1 + C (∇ΠN )(v(x)) ≤ (1 + CΛR) where Λ = ∇ΠN + BR (0) L∞ (Nδ ) . |∇(u( + (0) BR + L2 (BR (0)) + BR (0) |∇v|2 2 Rx ) ∇ u( |x| + CΛRn−1 , By direct calculations, we have Rx 2 ))| = |x| 1 d R n − 2 dR − 1 n−2 + BR (0) |∇u|2 + ∂BR (0)∩{xn >0} | ∂u 2 | dH n−1 .

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