We study Thurston’s Lipschitz and curve metrics, as well as the arc metric on the Teichmüller space of one-hold tori equipped with complete hyperbolic metrics with boundary holonomy of fixed length. We construct natural Lipschitz maps between two surfaces equipped with such hyperbolic metrics that generalize Thurston’s stretch maps and prove the following: (1) On the Teichmüller space of the torus with one boundary component, the Lipschitz and the curve metrics coincide and define a geodesic metric on this space. (2) On the same space, the arc and the curve metrics coincide when the length of the boundary component is \(\le 4{\text {arcsinh}}(1)\), but differ when the boundary length is large. We further apply our stretch map generalization to construct novel Thurston geodesics on the Teichmüller spaces of closed hyperbolic surfaces, and use these geodesics to show that the sum-symmetrization of the Thurston metric fails to exhibit Gromov hyperbolicity.

我们研究了Thurston的Lipschitz和曲线度量，以及一站式Tori的Teichmüller空间上的弧度量，该Toichmüller空间配备了具有固定长度边界完整性的完整双曲度量。我们在两个曲面之间构造自然的Lipschitz贴图，这些曲面配备了可推论Thurston拉伸图的双曲度量，并证明了以下几点：（1）在具有一个边界分量的圆环的Teichmüller空间上，Lipschitz和曲线度量重合并定义了测地线在这个空间上。（2）在同一空间上，当边界分量的长度为\（\ le 4 {\ text {arcsinh}}（1）\）时，圆弧和曲线度量重合，但当边界长度较大时会有所不同。我们进一步将拉伸图泛化应用到封闭双曲表面的Teichmüller空间上构造新颖的Thurston测地线，并使用这些测地线来证明Thurston度量的求和化未能表现出Gromov双曲线性。