Hessian of Hausdorff dimension on purely imaginary directions

Abstract: We extend classical results of Bridgeman-Taylor and McMullen on the Hessian of the Hausdorff dimension on quasi-Fuchsian space to the class of (1,1,2)-hyperconvex representations, a class introduced in arXiv:1902.01303 which includes small complex deformations of Hitchin representations and of $\Theta$-positive representations. We also prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion $\Gamma\to PO(n,1) \to PU(n,1)$ is positive definite when $\Gamma$ is co-compact in $PO(n,1)$ (unless $n=2$ and the deformation is tangent to $\mathfrak{X}(\Gamma,PO(2,1))).$