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

Abstract

We first extend the construction of the pressure metric to the deformation space of globally hyperbolic maximal Cauchy-compact anti-de Sitter structures. We show that, in contrast with the case of the Hitchin components, the pressure metric is degenerate and we characterize its degenerate locus. We then introduce a nowhere degenerate Riemannian metric adapting the work of Qiongling Li on the $$\mathrm {SL}(3,\mathbb {R})$$ -Hitchin component to this moduli space. We prove that the Fuchsian locus is a totally geodesic copy of Teichmuller space endowed with a multiple of the Weil–Petersson metric.