Convexity on Complex Hyperbolic Space

TL;DR

This paper investigates the properties of convex domains in complex hyperbolic space, providing bounds on volume-to-surface ratios and establishing the existence of certain convex domains based on curvature conditions.

Contribution

It improves existing bounds on volume-to-surface ratios for convex domains in complex hyperbolic space and proves the existence of $\lambda$-convex domains under specific curvature constraints.

Findings

Improved bounds for volume-to-surface ratios in complex hyperbolic space.

Existence of $\lambda$-convex domains of arbitrary radius if $\lambda \\leq k$.

Characterization of convex domains in complex hyperbolic space.

Abstract

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 $\lambda$. 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 $\CH$, a Hadamard manifold with constant holomorphic curvature equal to $-4k^2$. Furthermore, we give some specific properties of convex domains in $\CH$ and we prove that $\lambda$-convex domains of arbitrary radius exists if $\lambda\leq k$.