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Journal of Convex Analysis 20 (2013), No. 2, 329--338
Copyright Heldermann Verlag 2013



Convexity on Complex Hyperbolic Space

Judit Abardia
Dep. de Matemàtiques, Facultat de Ciències, Universitat Auṭnoma, 08193--Bellaterra / Barcelona, Spain
juditab@mat.uab.cat

Eduardo Gallego
Dep. de Matemàtiques, Facultat de Ciències, Universitat Auṭnoma, 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

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