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Moduli space of Brody curves, energy and mean dimension. (English) Zbl 1168.32016
A Brody curve is a holomorphic map from the complex plane \(\mathbb C\) to a Hermitian manifold with bounded derivative.
In this paper, the author studies the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and the author studies its mean dimension which was introduced by M. Gromov in 1999. In this paper, the author introduces the notion of mean energy:
\[ e(f)=\limsup_{r\rightarrow\infty}\frac{2}{\pi r^{2}}T(r,f) \]
where \(T(r,f)\) is the Shimizu-Ahlfors characteristic function, and shows that this can be used to estimate the mean dimension. Actually the author shows \(\dim(M(X):\mathbb C)\leq 4e(X)\dim_{\mathbb C}X\). Applying the above relation to the case of \(X=\mathbb C\mathbb{P}^{N}\), the author shows a certain upper bound for \(\dim(M(\mathbb C\mathbb{P}^{N}):\mathbb C)\) but he can not decide whether his estimate is better than Gromov’s estimate in 1999 or not. The author thinks that the use of mean energy makes the related estimates sharper.
Reviewer: Pei-Chu Hu (Jinan)

MSC:
32H30 Value distribution theory in higher dimensions
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