Convexity on complex hyperbolic space (Q2841069)

A regular convex domain in a Riemannian manifold \(M\) is called \(\lambda\)-convex if its normal curvature at each point is greater than or equal to \(\lambda>0\). Relation between area and volume for expanding families of \(\lambda\)-convex domains in an Hadamard manifold \(M\) was studied in [\textit{A. A. Borisenko, E. Gallego} and \textit{A. Reventós}, Differ. Geom. Appl. 14, No. 3, 267--280 (2001; Zbl 0981.52006)]. In the present paper, the authors specialize and sharpen the main results of that work for the case when \(M\) is the complex hyperbolic space \(\mathbb{C}H^n(-4k^2)\), an Hadamard manifold of real dimension \(2n\) with sectional curvature \(K\) satisfying \(-4k^2 \leq K \leq -k^2\).NEWLINENEWLINEThe main results are the following two theorems.NEWLINENEWLINE{Theorem 1.1.} In \(\mathbb{C}H^n(-4k^2)\), if a family of compact \(\lambda\)-convex domains expands over the whole space then \(\lambda \leq k\).NEWLINENEWLINE{Theorem 1.2.} Let \(\{\Omega_t\}_{t\in\mathbb{R}^+}\) be a piecewise \(C^2\) family of \(\lambda\)-convex compact domains, \(0 \leq \lambda \leq k\), expanding over the whole space \(\mathbb{C}H^n(-4k^2)\), \(n \geq 2\). Then NEWLINE\[NEWLINE\frac{\lambda}{4nk^2} \leq \liminf_{t \to \infty} \frac{vol(\Omega_t)}{vol(\partial\Omega_t)} \leq \limsup_{t \to \infty} \frac{vol(\Omega_t)}{vol(\partial\Omega_t)} \leq \frac{1}{2nk}.NEWLINE\]NEWLINE Moreover, the upper bound is sharp.NEWLINENEWLINEAt the end of the paper, the specialization of the results to other non-compact rank one symmetric spaces is briefly considered.