Hessian of Hausdorff dimension on purely imaginary directions

Abstract

Bridgeman–Taylor (Math. Ann. 341 (2008), 927–943) and McMullen (Invent. Math. 173 (2008), 365–425) showed that the Weil–Petersson metric on Teichmüller space can be realized by looking at the infinitesimal change of the Hausdorff dimension of certain quasi-Fuchsian deformations. In this article, we give a similar geometric interpretation of the spectral gap pressure metric introduced by Bridgeman–Canary–Labourie–Sambarino (Geom. Dedicata 192 (2018), 57–86) on the Hitchin component for PSL d ( R ) ${{\mathsf {PSL}}}_d(\mathbb {R})$ . More generally, we investigate the Hessian of the Hausdorff dimension as a function on the space of (1,1,2)-hyperconvex representations, a class introduced in (J. reine angew. Math. 774 (2021), 1–51) which includes small complex deformations of Hitchin representations and of Θ $\Theta$ -positive representations. As another application, we prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion Γ → PO ( n , 1 ) → PU ( n , 1 ) ${{\Gamma }}\rightarrow {{\mathsf {PO}}}(n,1)\rightarrow {{\mathsf {PU}}}(n,1)$ is positive definite when Γ ${{\Gamma }}$ is co-compact in PO ( n , 1 ) ${{\mathsf {PO}}}(n,1)$ (unless n = 2 $n=2$ and the deformation is tangent to X ( Γ , PO ( 2 , 1 ) ) $\mathfrak {X}({{\Gamma }}, {{\mathsf {PO}}}(2,1))$ ).