We find a spinorial representation of a Riemannian or Lorentzian surface in a Lorentzian homogeneous space of dimension 3. We in particular obtain a representation theorem for surfaces in \(\mathbb {L}(\kappa ,\tau )\) spaces. We then recover the Calabi correspondence between minimal surfaces in \(\mathbb {R}^3\) and maximal surfaces in \(\mathbb {R}_1^3\), and obtain a new Lawson type correspondence between CMC surfaces in \(\mathbb {R}_1^3\) and in the 3-dimensional pseudo-hyperbolic space \(\mathbb {H}_1^{3}.\)

中文翻译：

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维数为 $$\varvec{3}$$ 3 的洛伦兹齐次空间中表面的旋涡表示

我们在维数为 3 的洛伦兹齐次空间中找到了黎曼或洛伦兹曲面的自旋表示。我们特别获得了\(\mathbb {L}(\kappa ,\tau )\)空间中的曲面的表示定理。然后我们恢复\(\mathbb {R}^3\)中的最小曲面和 \(\mathbb {R}_1^3\)中的最大曲面之间的 Calabi 对应关系，并获得\(\mathbb {R}_1^3\)中的 CMC 曲面之间的新 Lawson 类型对应关系。 (\mathbb {R}_1^3\)和在 3 维伪双曲空间\(\mathbb {H}_1^{3}.\)