Convexity on Complex Hyperbolic Space

In a Riemannian manifold a regular convex domain is said to be $\lambda$-convex if its normal curvature at each point is greater than or equal to $\lambda>0$. In a Hadamard manifold, the asymptotic behaviour of the quotient $\vol(\Omega_{t})/\vol(\partial\Omega_{t})$ for a family of $\lambda$-convex domains $\Omega_{t}$ expanding over the whole space has been studied and general bounds for this quotient are known. In this paper we improve this general result in the complex hyperbolic space $\mathbb{C}H^n(-4k^2)$, a Hadamard manifold with constant holomorphic curvature equal to $-4k^2$. Furthermore, we give some specific properties of convex domains in $\mathbb{C}H^n(-4k^2)$ and we prove that $\lambda$-convex domains of arbitrary diameter exists if $\lambda\leq k$.