Moduli Space of Brody Curves, Energy and Mean Dimension

We study the mean dimension of the moduli space of Brody curves. We introduce the notion of “mean energy” and show that this can be used to estimate the mean dimension. 1. Main results 1.1. Moduli space of Brody curves. M. Gromov introduced a remarkable notion of mean dimension in [6] (see also Lindenstrauss-Weiss [8] and Lindenstrauss [7]). In this paper we study the mean dimension of the moduli space of Brody curves. We introduce the notion of mean energy of Brody curves and study the relation between mean energy and mean dimension. Mean energy is, in some sense, an infinite dimensional version of characteristic number, and our approach is an attempt to attack an infinite dimensional index problem. Let CP be the complex projective space and [z0 : z1 : · · · : zN ] be the homogeneous coordinate in CP . We define the Fubini-Study metric form ωFS on CP N by (1) ωFS := √ −1 2π ∂∂̄ log (