Hessian of Hausdorff dimension on purely imaginary directions

TL;DR

This paper generalizes the understanding of how the Hausdorff dimension's second derivative behaves in complex deformations of certain hyperbolic representations, extending classical results to new classes.

Contribution

It extends classical results on the Hessian of Hausdorff dimension to (1,1,2)-hyperconvex representations, including small complex deformations of Hitchin and $ heta$-positive representations.

Findings

Hessian of Hausdorff dimension is positive definite in certain co-compact cases.

Extension of classical results to new classes of hyperconvex representations.

Identification of conditions where the Hessian is not positive definite.

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 $\mathfrak{X}(\Gamma,PO(2,1))).$