By Frederic Hélein, R. Moser
This publication intends to offer an advent to harmonic maps among a floor and a symmetric manifold and incessant suggest curvature surfaces as thoroughly integrable platforms. The presentation is on the market to undergraduate and graduate scholars in arithmetic yet can also be priceless 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 idea, for the 1st time, from the viewpoint of a differential geometer. crucial effects are uncovered with whole proofs (except for the final chapters, which require a minimum wisdom from the reader). a few proofs were thoroughly rewritten with the target, particularly, to explain the relation among finite suggest curvature tori, Wente tori and the loop workforce technique - a side mostly missed within the literature. The ebook is helping the reader to entry the guidelines of the speculation and to obtain a unified viewpoint of the topic.
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Extra resources for Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Lectures in Mathematics. ETH Zürich)
Thus u is weakly conformal. 1 Any harmonic map u: S2 -+ N is weakly conformal. Using this we can prove the following. 4. 1 Any harmonic map u: S2 -+S2 is either i) constant, or ii) holomorphic, or iii) antiholomorphic. The proof of this Corollary uses the following lemma. We prove both results in the appendix, at the end of the chapter. 1 Let U, V C C be two open subsets of C and u: U - V a weakly conformal function of class C2. If U is connected, then u is either a constant, or holomorphic, or antiholomorphic.
There is no simple formula similar to Weierstrass representation for this case. However, the following may be done instead. Replacing (a - ib) by A-2(a - ib) means replacing f by A-2f. Keeping H and w fixed, this gives rise to a deformation & of A. But... 3, since we have not changed If 1. Hence also in this case we can find a conjugate family of CMC immersion, coming from this A,,. However, the Gauss map needn't stay the same. This is a result due to 0. Bonnet . We are going to look at this construction a little more closely.
Zi 2 0 0 A+A-t = A 1 0 0 1 and set GA := FARA, and aA := Ra 1HARA. 7) transforms into dGA = GAaA. Moreover, the following can be computed: aA = 0 0 A_i -e" - e-"f 0 0 ie" - ie e"+e-"f -ie"+ie-"f 4 +*dw A +4 0 -1 0 1 0 0 0 0 0 0 0 0 0 e"+e-"f ie" - ie-" f f dz 0 -e" - e-" f -ie" + ie-" f d2. 0 Setting A-lai, ao, and Aa' the first, second and third of the terms on the right hand side, respectively, this gives the splitting ax = A-lai + ao + Aai, 5. The Gauss-Codazzi condition 48 where ai = a'. We observe that the matrix ao has non-vanishing entries according to the pattern 0 * 0 * 0 0 0 0 0 0 0 0 0 * * * 0 , and a', ai according to * In the particular case where f = -1, al takes the form 0 0 0 0 sinhw -i cosh w 1= 2 - sinhw i cosh w 0 dz, and al the conjugate of this.
Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Lectures in Mathematics. ETH Zürich) by Frederic Hélein, R. Moser