A solvable string on a Lorentzian surface

作者：Clelland Jeanne N; Vassiliou Peter J^*

来源：Differential Geometry and its Applications, 2014, 33: 177-198.

DOI：10.1016/j.difgeo.2013.10.009

• Pseudo-Riemannian surface
• Harmonic map
• Cauchy problem
• Weierstrass representation
• Darboux integrability

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 R-1,R-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, lambda. Despite the completeness of lambda 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 R-1,R-1 into lambda.

• 出版日期2014-3