By Frederic Hélein, R. Moser
This e-book intends to offer an creation to harmonic maps among a floor and a symmetric manifold and relentless suggest curvature surfaces as thoroughly integrable platforms. The presentation is on the market to undergraduate and graduate scholars in arithmetic yet can be priceless to researchers. it really is one of the first textbooks approximately integrable structures, their interaction with harmonic maps and using loop teams, and it provides the speculation, for the 1st time, from the perspective 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, specifically, to elucidate the relation among finite suggest curvature tori, Wente tori and the loop team process - a side principally ignored within the literature. The publication is helping the reader to entry the information of the idea and to obtain a unified standpoint of the topic.
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Extra resources for Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Lectures in Mathematics. ETH Zürich)
As z1 - oo, we get lim fl (z1) = 0. z,-+00 But recall that f1 is holomorphic. Liouville's theorem therefore implies that f = 0. 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.
2. It is now easy to see that X is a CMC immersion with mean curvature H. 6) is a necessary and sufficient condition for w, H, and f to belong to a CMC immersion. 6) takes the form Ow + sinh w cosh w = 0. The corresponding second fundamental form is then II=e2m sinhw 0 1\ 0 cosh w Conjugate minimal surfaces and conjugate CMC surfaces Recall Weierstrass representation X(z) = Re J JJJza ''-(v2 - 1) h 2(v2 + 1) (S)dK iv for a minimal surface. We have seen in Chapter 2 that if u is the Gauss map of X and P a stereographic projection, then we may choose v = P o (-u) and h- (a +ib)(1 + Iv12) 82* 5.
Let u: 12 - N be any map. Define the function (,,U I2 f= _ i_ Iz=l Iz - I -2i(', )=4(5 )z. Recall that f is holomor/phic if u is harmonic. Since this property of u is invariant by conformal changes of variables, one might be interested in the behaviour of f by such a transformation. So let's choose a conformal map ¢:122 -12,, where 121 is the domain of a map u: 12, -, N, and let's check how the corresponding function f transforms. We write z2 and z, = O(z2) for the coordinates of 122 and 12,, respectively.
Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems (Lectures in Mathematics. ETH Zürich) by Frederic Hélein, R. Moser