Convexity on Complex Hyperbolic Space

In a Riemannian manifold a regular convex domain is said to be

-convex if its normal curvature at each point is greater than or equal to

\lambda>0 λ > 0

. In a Hadamard manifold, the asymptotic behaviour of the quotient

\mathop{\rm vol}(\Omega_{t})/\mathop{\rm vol}(\partial\Omega_{t}) v o l ( Ω t ) / v o l ( ∂ Ω t )

for a family of

\lambda λ

-convex domains

\Omega_{t} 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

\mathbb{C}H^n(-4k^2) C H n ( − 4 k 2 )

, a Hadamard manifold with constant holomorphic curvature equal to

-4k^2 − 4 k 2

. Furthermore, we give some specific properties of convex domains in

\mathbb{C}H^n(-4k^2) C H n ( − 4 k 2 )

and we prove that

\lambda λ

-convex domains of arbitrary diameter exists if

\lambda\leq k λ ≤ k

.