Boundary behavior of a infinitesimal metric and intrinsic measure on domains and moduli space

We investigate the intrinsic volume (n, n)-form of Eisenman and Kobayashi and the complex Finder metric of Kobayashi and Royden. We show that the natural logarithm of this measure and metric satisfy a modified sub-mean value property near the boundary of particular bounded domains. Then we illustrate how a bounded strictly pseudoconvex domain is such a domain which exhibits this boundary behavior for the Kobayashi-Royden metric. Also, we show how the moduli space of Riemann surfaces of given genus larger than one is such a particular domain which exhibits this boundary behavior for the Eisenman-Kobayashi volume form. Last, we show the equivalence of the Caratheodory and Eisenman-Kobayashi volume forms on Teichmuller space of Riemann surfaces of given genus larger than one.