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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.

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
Full Text: DOI arXiv
[1] McDuff, null (1994)
[2] DOI: 10.1007/BF02810577 · Zbl 0978.54026 · doi:10.1007/BF02810577
[3] DOI: 10.1007/BF02698858 · Zbl 0978.54027 · doi:10.1007/BF02698858
[4] Kollár, Birational Geometry of Algebraic Varieties (1998) · Zbl 0926.14003 · doi:10.1017/CBO9780511662560
[5] Tsukamoto, J. Math. Kyoto Univ. 48 pp 445– (2008)
[6] Gromov, J. Differential Geom. 18 pp 1– (1983)
[7] Gilbarg, Elliptic Partial Differential Equations of Second Order (2001)
[8] DOI: 10.2307/1998216 · Zbl 0416.32013 · doi:10.2307/1998216
[9] DOI: 10.2307/2661392 · Zbl 0970.30016 · doi:10.2307/2661392
[10] DOI: 10.1023/A:1009841100168 · Zbl 1160.37322 · doi:10.1023/A:1009841100168
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