On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 393-417.

Let Ω 1 and Ω 2 be strongly pseudoconvex domains in n and f:Ω 1 Ω 2 an isometry for the Kobayashi or Carathéodory metrics. Suppose that f extends as a C 1 map to Ω ¯ 1 . We then prove that f| Ω 1 :Ω 1 Ω 2 is a CR or anti-CR diffeomorphism. It follows that Ω 1 and Ω 2 must be biholomorphic or anti-biholomorphic.

Classification : 32F45, 32Q45
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Seshadri, Harish. On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 3, pp. 393-417. http://archive.numdam.org/item/ASNSP_2006_5_5_3_393_0/

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