洛伦兹曲面上的可解弦
A solvable string on a Lorentzian surface
摘要：
证明了除了Vessiot群是半单李群的中心扩张的非线性sigma模型外，还存在Darboux可积且具有可解Vessiot群的非线性sigma模型.它们支配Minkowski空间$\mathbb{R}^{1,1}$和某些完整的、非常曲率2-度量之间的调和映射。Vessiot群的可解性允许将一般柯西问题简化为求积。我们将Minkowski空间中调和映射的特殊情况处理为非常曲率Lorentzian 2-度量$\boldsymbolλ$。尽管$\boldsymbolλ$的完整性，我们表现出一个柯西问题，真正的分析初始数据，在有限时间内爆炸。对于从$\mathbb{R}^{1,1}$到$\boldsymbolλ$的所有调和映射，我们也导出了一个双曲Weierstrass表示公式.
英文摘要：
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 $\mathbb{R}^{1,1}$ 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, $\boldsymbolλ$. Despite the completeness of $\boldsymbolλ$ 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 $\mathbb{R}^{1,1}$ into $\boldsymbolλ$.
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