### Abstract

It is shown that there are nonlinear sigma models which are Darboux integrable and possess a solvable Vessiot group in addition to those whose Vessiot groups are central extensions of semi-simple Lie groups. They govern harmonic maps between Minkowski space R1,1 and certain complete, non-constant curvature 2-metrics. The solvability of the Vessiot group permits a reduction of the general Cauchy problem to quadrature. We treat the specific case of harmonic maps from Minkowski space into a non-constant curvature Lorentzian 2-metric, λ. Despite the completeness of λ we exhibit a Cauchy problem with real analytic initial data which blows up in finite time. We also derive a hyperbolic Weierstrass representation formula for all harmonic maps from R1,1 into λ.

Original language | English |
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Pages (from-to) | 177-198 |

Number of pages | 22 |

Journal | Differential Geometry and Its Applications |

Volume | 33 |

DOIs | |

Publication status | Published - Mar 2014 |

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*Differential Geometry and Its Applications*,

*33*, 177-198. https://doi.org/10.1016/j.difgeo.2013.10.009

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*Differential Geometry and Its Applications*, vol. 33, pp. 177-198. https://doi.org/10.1016/j.difgeo.2013.10.009

**A solvable string on a Lorentzian surface.** / Clelland, Jeanne; VASSILIOU, Peter.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A solvable string on a Lorentzian surface

AU - Clelland, Jeanne

AU - VASSILIOU, Peter

PY - 2014/3

Y1 - 2014/3

N2 - It is shown that there are nonlinear sigma models which are Darboux integrable and possess a solvable Vessiot group in addition to those whose Vessiot groups are central extensions of semi-simple Lie groups. They govern harmonic maps between Minkowski space R1,1 and certain complete, non-constant curvature 2-metrics. The solvability of the Vessiot group permits a reduction of the general Cauchy problem to quadrature. We treat the specific case of harmonic maps from Minkowski space into a non-constant curvature Lorentzian 2-metric, λ. Despite the completeness of λ we exhibit a Cauchy problem with real analytic initial data which blows up in finite time. We also derive a hyperbolic Weierstrass representation formula for all harmonic maps from R1,1 into λ.

AB - It is shown that there are nonlinear sigma models which are Darboux integrable and possess a solvable Vessiot group in addition to those whose Vessiot groups are central extensions of semi-simple Lie groups. They govern harmonic maps between Minkowski space R1,1 and certain complete, non-constant curvature 2-metrics. The solvability of the Vessiot group permits a reduction of the general Cauchy problem to quadrature. We treat the specific case of harmonic maps from Minkowski space into a non-constant curvature Lorentzian 2-metric, λ. Despite the completeness of λ we exhibit a Cauchy problem with real analytic initial data which blows up in finite time. We also derive a hyperbolic Weierstrass representation formula for all harmonic maps from R1,1 into λ.

KW - Cauchy problem

KW - Darboux integrability

KW - Harmonic map

KW - Pseudo-Riemannian surface

KW - Weierstrass representation

U2 - 10.1016/j.difgeo.2013.10.009

DO - 10.1016/j.difgeo.2013.10.009

M3 - Article

VL - 33

SP - 177

EP - 198

JO - Differential Geometry and Its Applications

JF - Differential Geometry and Its Applications

SN - 0926-2245

ER -