Convexity on complex hyperbolic space

In a Riemannian manifold a regular convex domain is said to be λ-convex if its normal curvature at each point is greater than or equal to λ < 0. In a Hadamard manifold, the asymptotic behaviour of the quotient vol(Ω_t)/vol(∂Ω_t)for a family of λ-convex domains Ω_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 ℂH ⁿ(-4k²), a Hadamard manifold with constant holomorphic curvature equal to - 4². Furthermore, we give some specific properties of convex domains in ℂHⁿ(-4k²)and we prove that λ-convex domains of arbitrary diameter exists if λ ≤ k.