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
SN - 0926-2245
VL - 33
SP - 177
EP - 198
JO - Differential Geometry and Its Applications
JF - Differential Geometry and Its Applications
ER -