On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 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
@article{ASNSP_2006_5_5_3_393_0,
     author = {Seshadri, Harish},
     title = {On isometries of the carath\'eodory and {Kobayashi} metrics on strongly pseudoconvex domains},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {393--417},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {3},
     year = {2006},
     mrnumber = {2274785},
     zbl = {1170.32309},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2006_5_5_3_393_0/}
}
TY  - JOUR
AU  - Seshadri, Harish
TI  - On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2006
SP  - 393
EP  - 417
VL  - 5
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2006_5_5_3_393_0/
LA  - en
ID  - ASNSP_2006_5_5_3_393_0
ER  - 
%0 Journal Article
%A Seshadri, Harish
%T On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2006
%P 393-417
%V 5
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2006_5_5_3_393_0/
%G en
%F ASNSP_2006_5_5_3_393_0
Seshadri, Harish. On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 3, pp. 393-417. http://archive.numdam.org/item/ASNSP_2006_5_5_3_393_0/

[1] Z. M. Balogh and M. Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000), 504-533. | MR | Zbl

[2] Z. M. Balogh and M. Bonk, Pseudoconvexity and Gromov hyperbolicity, C. R. Acad. Sci. Paris Sèr. I Math. 328(1999), 597-602. | MR | Zbl

[3] F. Berteloot, Attraction des disques analytiques et continuitè höldérienne d'applications holomorphes propres, In: “Topics in Complex Analysis”(Warsaw, 1992), Banach Center Publ., Vol. 31, Polish Acad. Sci., Warsaw, 1995. | MR | Zbl

[4] J. Bland, T. Duchamp and M. Kalka, On the automorphism group of strictly convex domains in n , In: “Complex Differential Geometry and Nonlinear Differential Equations” (Brunswick, maine, 1984), Comtemp. Math. Vol. 49, 1986, 19-30. | MR | Zbl

[5] S. Y. Cheng and S. T. Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman's equation, Comm. Pure Appl. Math. 33 (1980), 507-544. | MR | Zbl

[6] K. Diederich and S. Pinchuk, Proper holomorphic maps in dimension 2 extend, Indiana Univ. Math. J. 44 (1995), 1089-1125. | MR | Zbl

[7] K. Diederich and S. Webster, A reflection principle for degenerate real hypersurfaces, Duke Math. J. 47 (1980), 835-843. | MR | Zbl

[8] F. Forstneric, An elementary proof of Fefferman's theorem, Expo. Math. 10 (1992), 135-149. | MR | Zbl

[9] H. Gaussier, K-T. Kim and S. G. Krantz, A note on the Wong-Rosay theorem in complex manifolds, Complex Variables Theory Appl. 47 (2002), 761-768. | MR | Zbl

[10] I. Graham, Boundary behavior of the Carathèodory and Kobayashi metrics on strongly pseudoconvex domains in n with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219-240. | MR | Zbl

[11] I. Graham, Holomorphic mappings into strictly convex domains which are Kobayashi isometries at one point, Proc. Amer. Math. Soc. 105 (1989), 917-921. | MR | Zbl

[12] R. E. Greene and S. G. Krantz, Deformation of complex structures, estimates for the ¯ equation, and stability of the Bergman kernel, Adv. Math. 43 (1982), 1-86. | MR | Zbl

[13] R. E. Greene and S. G. Krantz, Stability of the Carathèodory and Kobayashi metrics and applications to biholomorphic mappings, In: “Complex Analysis of Several Variables (Madison, Wis., 1982), 77-93, Proc. Sympos. Pure Math. 41, Amer. Math. Soc., Providence, RI, 1984. | MR | Zbl

[14] S. Kobayashi, Intrinsic metrics on complex manifolds, Bull. Amer. Math. Soc. 73 (1967), 347-349. | MR | Zbl

[15] L. D. Kay, On the Kobayashi-Royden metric for ellipsoids, Math. Ann. 289 (1991), 55-72. | MR | Zbl

[16] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. | Numdam | MR | Zbl

[17] D. Ma, On iterates of holomorphic maps, Math. Z. 207 (1991), 417-428. | MR | Zbl

[18] S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. Math. 40 (1939), 400-416. | MR | Zbl

[19] L. Nirenberg, S. Webster and P. Yang, Local boundary regularity of holomorphic mappings, Comm. Pure Appl. Math. 33 (1980), 305-338. | MR | Zbl

[20] G. Patrizio, On holomorphic maps between domains in n , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 13 (1986), 267-279. | Numdam | MR | Zbl

[21] S. I. Pinchuk, A boundary uniqueness theorem for holomorphic functions of several complex variables, Mat. Zametki 15 (1974), 205-212. | MR | Zbl

[22] S. I. Pinchuk, On proper holomorphic mappings of strictly pseudoconvex domains, Siberian Math. J. 15 (1974), 644-649. | Zbl

[23] S. I. Pinchuk, Holomorphic inequivalence of certain classes of domains in n , Mat. Sb. (N.S.) 111(153) (1980), 67-94. | MR | Zbl

[24] S. I. Pinchuk and S. V. Khasanov, Asymptotically holomorphic functions and their applications, Math. USSR-Sb. 62 (1992), 541-550. | MR | Zbl

[25] S. I. Pinchuk, The scaling method and holomorphic mappings In: “Several Complex Variables and Complex Geometry”, Part 1 (Santa Cruz, CA, 1989), 151-161, Proc. Sympos. Pure Math., 52, Part 1, Amer. Math. Soc., Providence, RI, 1991. | MR | Zbl

[26] J-P. Rosay, Sur une caractèrisation de la boule parmi les domaines de n par son groupe d’automorphismes, Ann. Inst. Fourier (Grenoble), 29 (1979), 91-97. | Numdam | MR | Zbl

[27] K. Verma, Boundary regularity of correspondences in 2 , Math. Z. 231 (1999), 253-299. | MR | Zbl

[28] E. Vesentini, Complex geodesics and holomorphic maps, In: Symposia Mathematica”, Vol. XXVI, Rome, 1980, 211-230, Academic Press, London-New York, 1982. | MR | Zbl

[29] J. P. Vigué, Caractérisation des automorphismes analytiques d'un domaine convexe borné, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 101-104. | Zbl

[30] S. Webster, On the reflection principle in several complex variables, Proc. Amer. Math. Soc. 71 (1978), 26-28. | MR | Zbl

[31] B. Wong, Characterization of the unit ball in n by its automorphism group, Invent. Math. 41 (1977), 253-257. | MR | Zbl

[32] J. Y. Yu, Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains, Trans. Amer. Math. Soc. 347 (1995), 587-614. | MR | Zbl