Finite type Lorentz harmonic maps and the method of Symes.

*(English)*Zbl 1027.58012There 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.

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.

Reviewer: Ian McIntosh (York)

PDF
BibTeX
Cite

\textit{J. Dorfmeister} and \textit{I. Sterling}, Differ. Geom. Appl. 17, No. 1, 43--53 (2002; Zbl 1027.58012)

Full Text:
DOI

##### References:

[1] | Burstall, F.E.; Pedit, F., Harmonic maps via adler – kostant – symes theory, (), 221-272 · Zbl 0828.58021 |

[2] | Burstall, F.E.; Pedit, F., Dressing orbits of harmonic maps, Duke math J., 80, 353-382, (1995) · Zbl 0966.58007 |

[3] | Dobriner, H., Die flächen constanter krümmung mit einem system sphärischer krümmungslinien dargestellt mit hilfe von theta functionen zweier variabeln, Acta math., 9, 73-104, (1886) · JFM 18.0425.01 |

[4] | Dorfmeister, J.; Pedit, F.; Wu, H., Weierstrass type representation of harmonic maps into symmetric spaces, Comm. anal. geom., 6, 633-668, (1998) · Zbl 0932.58018 |

[5] | Ercoloni, N., Generalized theta functions and homoclinic varieties, Proc. symp. pure math., 49, 1, 87-100, (1989) |

[6] | Ercoloni, N.M.; Forrest, M.G., The geometry of real sine-Gordon wavetrains, Comm. math. phys., 99, 1-49, (1985) · Zbl 0591.35065 |

[7] | Guest, M., Harmonic maps, loop groups, and integrable systems, (1997), Cambridge Univ. Press · Zbl 0898.58010 |

[8] | Kozel, V.A.; Kotlyarov, V.P., Quasi-periodic solutions of the equation Utt−uxx+sinu, Dokl. akad. nauk ukrain. SSR ser. A, 10, 878-881, (1976) · Zbl 0337.35003 |

[9] | V.B. Matveev, Abelian functions and solitons, Preprint No. 373, University of Wroclow, 1976 |

[10] | Melko, M.; Sterling, I., Applications of soliton theory to the construction of pseudospherical surfaces in \(R\^{}\{3\}\), Ann. global anal., 11, 65-107, (1993) · Zbl 0810.53003 |

[11] | Melko, M.; Sterling, I., Integrable systems, harmonic maps and the classical theory of surfaces, (), 129-144 · Zbl 0814.58014 |

[12] | Pinkall, U.; Sterling, I., On the classification of constant Mean curvature tori, Ann. of math., 130, 407-451, (1989) · Zbl 0683.53053 |

[13] | N. Schmidt, www.gang.umass.edu/ nick |

[14] | N. Schmidt, private communication |

[15] | C.-L. Terng, K. Uhlenbeck, Bäcklund transformation and loop group actions, Comm. Pure. Appl. Math., to appear |

[16] | M. Toda, Pseudospherical surfaces via moving frames and loop groups, Dissertation, Univ. of Kansas, May 2000 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.