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arXiv · 2012.13701

Slice rigidity property of holomorphic maps Kobayashi-isometrically preserving complex geodesics

Filippo Bracci·Łukasz Kosiński·Włodzimierz Zwonek
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Abstract

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 $F:M\to N$ 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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Filippo Bracci, Łukasz Kosiński, Włodzimierz Zwonek. 2020-12-26. Slice rigidity property of holomorphic maps Kobayashi-isometrically preserving complex geodesics. https://arxiv.org/abs/2012.13701

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