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A new Schwarz-Pick Lemma at the boundary and rigidity of holomorphic maps

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 Added by Filippo Bracci
 Publication date 2020
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and research's language is English




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147 - Xieping Wang , Guangbin Ren 2015
In this paper, we generalize a recent work of Liu et al. from the open unit ball $mathbb B^n$ to more general bounded strongly pseudoconvex domains with $C^2$ boundary. It turns out that part of the main result in this paper is in some certain sense just a part of results in a work of Bracci and Zaitsev. However, the proofs are significantly different: the argument in this paper involves a simple growth estimate for the Caratheodory metric near the boundary of $C^2$ domains and the well-known Grahams estimate on the boundary behavior of the Caratheodory metric on strongly pseudoconvex domains, while Bracci and Zaitsev use other arguments.
154 - Guangbin Ren , Xieping Wang 2015
In this paper, we present an alternative and elementary proof of a sharp version of the classical boundary Schwarz lemma by Frolova et al. with initial proof via analytic semigroup approach and Julia-Caratheodory theorem for univalent holomorphic self-mappings of the open unit disk $mathbb Dsubset mathbb C$. Our approach has its extra advantage to get the extremal functions of the inequality in the boundary Schwarz lemma.
170 - Mark Elin , David Shoikhet 2011
In this paper we give some quantative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy--Wolff point. This enable us to establish an asymptotic rigidity property for semigroups of parabolic type.
In this paper we study the following slice rigidity property: given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $mathcal F$ of $M$, when is it true that every holomorphic map $F:Mto N$ which maps isometrically every complex geodesic of $mathcal F$ onto a complex geodesic of $N$ is a biholomorphism? Among other things, we prove that this is the case if $M, N$ are smooth bounded strictly (linearly) convex domains, every element of $mathcal F$ contains a given point of $overline{M}$ and $mathcal F$ spans all of $M$. More general results are provided in dimension $2$ and for the unit ball.
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