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Deformation of Brody curves and mean dimension. (English) Zbl 1186.32002
A holomorphic curve \(f:{\mathbb C}\to {\mathbb C}\mathbb{P}^N\) is called a Brody curve if its pointwise norm (with respect to the Fubini-Study metric) satisfies the inequality \(|df|\leq 1\). Let \({\mathcal M}({\mathbb C}\mathbb{P}^N)\) be the space of Brody curves with the compact-open topology. The author studies the “mean dimension” \(\text{dim}({\mathcal M}({\mathbb C}\mathbb{P}^N):{\mathbb C})\) and estimates it from above and below in terms of the so-called “mean energy.” For that purpose, a new deformation theory of Brody curves is developed.

MSC:
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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