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On harmonic maps from \(R^{1,1}\) into Hilbert loop groups. (English) Zbl 0871.58024
Let \(H_s(LG)=\{\gamma:S^1\to G\mid \gamma\) is \(H_s\) integrable} \((s>\frac12)\) be the real Hilbert loop group of a compact Lie group. The authors prove that the Cauchy problem for harmonic maps from the Minkowski plane \(\mathbb{R}^{1,1}\) into Hilbert loop groups \(H_s(LG)\) exists globally and uniquely when \(s>\frac34\). The construction of the harmonic maps strongly depends on some geometric properties of the loop groups.

MSC:
58E20 Harmonic maps, etc.
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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References:
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[2] DOI: 10.1002/cpa.3160330604 · Zbl 0475.58005 · doi:10.1002/cpa.3160330604
[3] DOI: 10.1016/0003-4916(82)90077-X · Zbl 0512.58018 · doi:10.1016/0003-4916(82)90077-X
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