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Finite type Lorentz harmonic maps and the method of Symes. (English) Zbl 1027.58012
There is a general principle in the theory of Lax equations, which was observed by W. W. Symes [Invent. Math. 59, 13-51 (1980; Zbl 0474.58009)] (and also independently by Reyman and Semenov-Tian-Shansky over several papers published around the same time), which can be summarised as follows. In a Lie algebra \(\mathfrak g\) which splits into a (vector space) direct sum of subalgebras \({\mathfrak g}_++{\mathfrak g}_-\) the unique solution \(L:\mathbb{R}\to{\mathfrak g}\) of the Lax equation \[ dL/dt = [L,L_+],\;L(0)=L_0 \] (where \(L_+\) denotes projection onto \({\mathfrak g}_+\)) is given by \(L=\text{ Ad}g_+^{-1}.L_0\) whenever \(\exp(tL_0)\) factorises into the product \(g_+g_-\) with \(g_{\pm}\) lying in the Lie subgroups Lie(\({\mathfrak g}_\pm\)) of the Lie group Lie(\({\mathfrak g}\)). As stated here, this is a simple algebraic calculation of about four lines. The only significant issue is the existence of the factorisation: in most cases this is only possible on an open dense subset of Lie(\({\mathfrak g}\)), whence the solution \(L\) is really only locally defined.
This result found an application to the construction of harmonic maps of the 2-plane into compact symmetric (or \(k\)-symmetric) spaces in the work of F. E. Burstall and F. Pedit [in Aspects Math. E23, 221-272 (1994; Zbl 0828.58021)] where the Lie algebra is a loop algebra and the appropriate splitting could be chosen to make the factorisation global. Those harmonic maps which arose from the Lax equation were dubbed “finite type”, and it followed that all such were also obtainable by the Symes construction (and vice versa).
The article under review describes which loop algebra and splitting to choose to construct Lorentz harmonic maps \(\mathbb{R}^{1,1}\to S^2\). In this case the factorisation only applies to an open dense subset of the loop group, so that the maps which result are only locally defined. The authors verify that Symes’ observation still holds, by doing a direct calculation.

58E20 Harmonic maps, etc.
53C43 Differential geometric aspects of harmonic maps
Full Text: DOI
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