By Frederic Hélein, R. Moser
This ebook intends to provide an creation to harmonic maps among a floor and a symmetric manifold and incessant suggest curvature surfaces as thoroughly integrable structures. The presentation is available to undergraduate and graduate scholars in arithmetic yet can also be invaluable to researchers. it really is one of the first textbooks approximately integrable platforms, their interaction with harmonic maps and using loop teams, and it offers the speculation, for the 1st time, from the perspective of a differential geometer. an important effects are uncovered with entire proofs (except for the final chapters, which require a minimum wisdom from the reader). a few proofs were thoroughly rewritten with the target, specifically, to explain the relation among finite suggest curvature tori, Wente tori and the loop staff procedure - a side principally missed within the literature. The ebook is helping the reader to entry the tips of the idea and to procure a unified standpoint of the topic.
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Extra info for Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Lectures in Mathematics. ETH Zürich)
Proof. We build a perturbation of u which is a modulation of the action of X by a function ¢ E C' (SZ, R). It gives us a map v, from St to R" such that v,(x + s$(x)X(x)) = u(x). Note that if s is sufficiently small, x 1 1 x + s4(x)X(x) is a diffeomorphism of Q. ;)X (l;))) d6 + o(s). [Il - s¢(x)d(X (x))]) +s J [- E E 0,(3=1 i=1 m +E 0=1 a a a (x) [1 + s¢(x)divX (x)] dx (x)X s(x) OU, (x, u(x), du(x)) ° (x)Xc(x)L(x, u(x), du(x))] dx + o(s) 3. Variational point of view and Noether's theorem An [u] - s E J a,p=1 in 33 aX0 (x)X' (x)H; (x) dx + o(s).
Step 1: Local existence. (x)) aM' where we denote (MA. (x)) aM :_ E M,Ao(x)kaj . 1
0 = 0. 1 We say that u: S2 S" is isotropic, if and only if 17,,Q = 0 for all a,,3 > 0 satisfying a +,3 >- 1. 1. We will apply induction on a +,3. The claim is clear for a + Q = 1, since 7701 = 7710 = (au, u) = 0. Assume now that -y E N is given, and that dap = 0 for all a,,3 such that 1