The Bloch constant of holomorphic mappings in several complex variables (Q1206080)

Let \(\Omega\subset\mathbb{C}^ n\) be a bounded homogeneous domain with \(0\in\Omega\). For \(1\leq K<+\infty\) put \[ B_ K(\Omega):=\{F\in{\mathcal O}(\Omega,\mathbb{C}^ n):\;\text{det} F'(0)=1,\;\forall\varphi\in\text{Aut}(\Omega):\;\|(F\circ\varphi)'(0)\|\leq K\}. \] The authors announce (without proofs) several results characterizing the Bloch constant \[ B:=\inf_{F\in B_ K(\Omega)}\sup\{r(a,F):a\in\Omega\}, \] where \(r(a,F)\) denotes the supremum of all \(r\geq 0\) for which there exists a domain \(R\subset\Omega\), \(a\in R\), such that \(F\) maps biholomorphically \(R\) onto the ball \(B(F(a),r)\).