Trois problèmes géométriques d'hyperbolicité complexe et presque complexe

Abstract : This thesis is dedicated to the study of three problems of complex and almost complex hyperbolicity. Its first part is dedicated to the research of a quantitative consequence to Kobayashi hyperbolicity, which is a qualitative property. The result we obtain has the form of an isoperimetric inequality that suggests Ahlfors' inequality, the central result of the theory of covering surfaces. Its proof uses only riemannian tools.The second part of the thesis is dedicated to the proof of an almost complex version of Borel's theorem, which says that an entire curve in the compex preojective plane missing four lines in general position is degenerate. In an almost compex context, we can obtain a similar result for entire J-curves just by replacing projective lines by J-lines. The proof of this result uses central projections and Ahlfors' theory of covering surfaces.The last part is dedicated to the proof of an almost complex version of Bloch's theorem, which says that given a sequence of holomorphic discs in the projective plane, either it is normal, either it converges in some sens to a reunion of three lines. Our result will show in particular that the complementary set of four J-lines in general position is hyperbolic modulo three J-lines.
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Benoît Saleur. Trois problèmes géométriques d'hyperbolicité complexe et presque complexe. Mathématiques générales [math.GM]. Université Paris Sud - Paris XI, 2011. Français. ⟨NNT : 2011PA112256⟩. ⟨tel-00647337⟩

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