Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 20 (2013), No. 2, 329--338Copyright Heldermann Verlag 2013 Convexity on Complex Hyperbolic Space Judit Abardia Dep. de Matemàtiques, Facultat de Ciències, Universitat Autònoma, 08193--Bellaterra / Barcelona, Spain juditab@mat.uab.cat Eduardo Gallego Dep. de Matemàtiques, Facultat de Ciències, Universitat Autònoma, 08193--Bellaterra / Barcelona, Spain egallego@mat.uab.cat [Abstract-pdf] \newcommand{\vol}{\mathop{\rm vol}} 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.\par 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 $\lambda\leq k$. Keywords: Complex hyperbolic space, convex domain, volume, area. MSC: 52A20; 52A55 [ Fulltext-pdf  (132  KB)] for subscribers only.