Carathéodory extremal maps of ellipsoids (Q1127012)

Let \((M,p)\) and \((N,q)\) be pointed domains in \(\mathbb C^n\); a \textit{C-extremal map} from \((M,p)\) to \((N,q)\) is a holomorphic map \(f:M \to N\) with \(f(p) = q\) for which \(| \det f'(p)| \) is largest among all such maps. A \textit{generalized ellipsoid} in \(\mathbb C^n\) is a domain of the form \[ E(p,m) = \Biggl\{z \in \mathbb C^{p_1} \times \cdots \times \mathbb C^{p_k} : \sum_{j=1}^k \| z_j \|^{2m_j} < 1 \Biggr\} \] where \(p_1 + \cdots + p_k = n\) and \(0 < m_j \leq \infty\); here \(\| z_j \|\) denotes the euclidean norm of \(z_j\) in \(\mathbb C^{p_j}\) and an infinite value of \(m_j\) is understood as a limit. The author finds C-extremal maps in both directions between \((E(p,m),0)\) and \((E(p,l),0)\) provided \(m_j \geq l_j \geq 1\) for all \(j\) or \(1 \geq m_j \geq l_j\) for all \(j\). A direct computation then yields an explicit value for the distance between these pointed generalized ellipsoids in the space \(\mathcal T_n\) of (biholomorphy classes of) \(n\)-dimensional pointed taut complex manifolds, equipped with the author's extremal metric [Math. Ann. 292, No. 3, 533-546 (1992; Zbl 0738.32019)]. As an application, the author writes formulas for some geodesic segments in \(\mathcal T_n\), all points of which are (biholomorphy classes of) such ellipsoids; when \(n \geq 2\) there are arbitrarily close points in \(\mathcal T_n\) joined by infinitely many such geodesics. Finally he observes that \(\mathcal T_n\) contains isolated points, for example (biholomorphy classes of) the \(h\)-minimal manifolds of \textit{J. Winkelmann} [Int. J. Math. 8, No. 1, 149-168 (1997; Zbl 0880.32002)].