On regions biholomorphically equivalent to a ball (Q1177569)

The author proves some characterizations of the ball \(D\) in \(\mathbb{C}^ n\) in terms of the properties of the Kobayashi metric in \(D\). In particular, if \(D\) is a strictly pseudo-convex domain with simply connected \(C^ \infty\) boundary \(\partial D\), then \(D\) is biholomorhic to a ball if and only if the Kobayashi metric is a Hermitian metric near the boundary \(\partial D\). The results are generalisations of the characterization of the ball given by \textit{C. M. Stanton} [Math. Ann. 264, 271-275 (1983; Zbl 0501.32002)].