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Slice Rigidity Property of Holomorphic Maps Kobayashi-Isometrically Preserving Complex Geodesics

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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 \({\mathscr {F}}\) of M, when is it true that every holomorphic map \(F:M\rightarrow N\) which maps isometrically every complex geodesic of \({\mathscr {F}}\) 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 \({\mathscr {F}}\) contains a given point of \({\overline{M}}\) and \({\mathscr {F}}\) spans all of M. More general results are provided in dimension 2 and for the unit ball.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Filippo Bracci

  2. Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland

    Łukasz Kosiński & Włodzimierz Zwonek

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  1. Filippo Bracci
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  2. Łukasz Kosiński
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  3. Włodzimierz Zwonek
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F. Bracci: Partially supported by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics n.2017JZ2SW5, by INdAM and by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Ł Kosiński: Partially supported by NCN grant SONATA BIS No. 2017/26/E/ST1/00723 of the National Science Centre, Poland. W. Zwonek: Partially supported by the OPUS Grant No. 2015/17/B/ST1/00996 of the National Science Centre, Poland.

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Bracci, F., Kosiński, Ł. & Zwonek, W. Slice Rigidity Property of Holomorphic Maps Kobayashi-Isometrically Preserving Complex Geodesics. J Geom Anal 31, 11292–11311 (2021). https://doi.org/10.1007/s12220-021-00681-6

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  • DOI: https://doi.org/10.1007/s12220-021-00681-6

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