In this short note we explain how to adapt the construction of two Riemannian metrics on the $\mathrm{SL}(3,\mathbb{R})$-Hitchin component to the deformation space of globally hyperbolic anti-de Sitter structures: the pressure metric and the Loftin metric (studied by Qiongling Li). We show that the former is degenerate and we characterize its degenerate locus, whereas the latter is nowhere degenerate and the Fuchsian locus is a totally geodesic copy of Teichmuller space endowed with a multiple of the Weil-Petersson metric.

中文翻译：

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GHMC 反德西特结构模空间的黎曼度量

在这个简短的说明中，我们解释了如何使 $\mathrm{SL}(3,\mathbb{R})$-Hitchin 分量上的两个黎曼度量的构造适应全局双曲反德西特结构的变形空间：压力度量和 Loftin 度量（由李琼玲研究）。我们证明前者是退化的并且我们描述了它的退化轨迹，而后者在任何地方都没有退化并且 Fuchsian 轨迹是 Teichmuller 空间的完全测地线副本，具有 Weil-Petersson 度量的倍数。