Asymptotics of invariant metrics in the normal direction and a new characterisation of the unit disk

We give improvements of estimates of invariant metrics in the normal direction on strictly pseudoconvex domains. Specifically we will give the second term in the expansion of the metrics. This depends on an improved localisation result and estimates in the one variable case. Finally we will give a new characterisation of the unit disk in C in terms of the asymptotic behaviour of quotients of invariant metrics.