Riemannian metrics on the moduli space of GHMC anti-de Sitter structures

Abstract

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 Teichm\"uller space endowed with a multiple of the Weil-Petersson metric.