Brody curves and mean dimension

Deformation of Brody Curves and Mean Dimension

The main purpose of this paper is to show that ideas of deformation theory can be applied to “infinite dimensional geometry”. We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we pr...

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 modu...

Brody curves omitting hyperplanes

A Brody curve, a.k.a. normal curve, is a holomorphic map f from the complex line C to the complex projective space P such that the family of its translations {z 7→ f(z + a) : a ∈ C} is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n =

Ja n 20 09 Brody curves omitting hyperplanes

A Brody curve, a.k.a. normal curve, is a holomorphic map f from the complex line C to the complex projective space P such that the family of its translations {z 7→ f(z + a) : a ∈ C} is normal. We prove that Brody curves omitting n hyperplanes in general position have growth order at most one, normal type. This generalizes a result of Clunie and Hayman who proved it for n =

Hausdorff Dimension and mean porosity