By Marco Abate, John Erik Fornaess, Xiaojun Huang, Jean-Pierre Rosay, Alexander Tumanov (auth.), Dmitri Zaitsev, Giuseppe Zampieri (eds.)

ISBN-10: 3540223584

ISBN-13: 9783540223580

The geometry of actual submanifolds in advanced manifolds and the research in their mappings belong to the main complicated streams of latest arithmetic. during this quarter converge the ideas of varied and complicated mathematical fields akin to P.D.E.'s, boundary worth difficulties, prompted equations, analytic discs in symplectic areas, complicated dynamics. For the diversity of subject matters and the strangely stable interplaying of alternative examine instruments, those difficulties attracted the eye of a few the most effective mathematicians of those most modern 20 years. in addition they entered as a polished content material of a sophisticated schooling. during this experience the 5 lectures of this quantity supply a good cultural history whereas giving very deep insights of present study activity.

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The geometry of genuine submanifolds in advanced manifolds and the research in their mappings belong to the main complicated streams of latest arithmetic. during this region converge the suggestions of assorted and complicated mathematical fields similar to P. D. E. 's, boundary worth difficulties, prompted equations, analytic discs in symplectic areas, complicated dynamics.

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If f being β-Julia at x ∈ ∂D would imply f ◦ ϕx being β-Julia at 1, one could try to apply the classical Julia-Wolﬀ-Carath´eodory theorem to f ◦ ϕx . 7 Let D ⊂ Cn be a complete hyperbolic domain equipped with a projection device at x ∈ ∂D preserving horospheres. Then for any f ∈ Hol(D, ∆) which is β-Julia at x, the composition f ◦ ϕx is β-Julia at 1. In particular, (f ◦ ϕx ) has non-tangential limit βτ at 1, where τ ∈ ∂∆ is the K-limit of f at x. 38 Marco Abate Proof. Since f is β-Julia and the projection device preserves horospheres we have f ◦ ϕx E(1, R) ⊆ E(τ, βR) for all R > 0.

Furthermore, a T -good geometrical projection device preserving horospheres is automatically good. 30 Marco Abate Proof. First of all, we claim that ϕx K(1, M ) ⊂ KzU∩D (x, M ) 0 for all M > 1. Indeed, if ζ ∈ K(1, M ) we have ζ ∈ E 1, M 2 /R(ζ) , where R(ζ) satisﬁes 1 2 log R(ζ) = ω(0, ζ) = kU∩D z0 , ϕx (ζ) as usual. But then ϕx (ζ) ∈ EzU∩D x, M 2 /R(ζ) , which immediately implies 0 U∩D ϕx (ζ) ∈ Kz0 (x, M ), as claimed. Now take z ∈ T (x, M, δ). Then we have kU∩D z, w − kU∩D (z0 , w) + kU∩D (z0 , z) ≤ 2kU∩D z, px (z) + kU∩D (px (z), w) − kU∩D (z0 , w) + kU∩D z0 , px (z) < 2ω(0, δ) + kU∩D (px (z), w) − kU∩D (z0 , w) + kU∩D z0 , px (z) for all w ∈ U ∩ D.

By construction, ϕx K(1, M ) ⊂ T (x, M, δ) for all 0 < δ < 1. Furthermore, if γ ∈ R there always are M > 1 and δ ∈ (0, 1) such that γ(t) ∈ T (x, M, δ) for all t close enough to 1. 11 A geometrical projection device at x ∈ ∂D is T -good if for any γ ∈ R then there exist M = M (γ) > 1 and δ = δ(γ) > 0 such that lim kT (x,M,δ) γ(t), γx (t) = 0. t→1− Example 20. The canonical projection device is T -good in all convex domains of ﬁnite type ([AT2]). 12 Let D ⊂ Cn be a domain equipped with a T -good geometrical projection device at x ∈ ∂D.

### Real Methods in Complex and CR Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, June 30 - July 6, 2002 by Marco Abate, John Erik Fornaess, Xiaojun Huang, Jean-Pierre Rosay, Alexander Tumanov (auth.), Dmitri Zaitsev, Giuseppe Zampieri (eds.)

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