Holomorphic isometries from the Poincaré disk into bounded symmetric domains of rank at least two
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2205-2240.

We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit n-ball for n2. On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit n-ball into any irreducible bounded symmetric domain of rank 2 which is not biholomorphic to any type-IV domain. In particular, our study provides many new examples of holomorphic isometries from the Poincaré disk into irreducible bounded symmetric domains of rank at least 2 except for type-IV domains.

Nous étudions d’abord les isométries holomorphes du disque de Poincaré dans le produit du disque unité et de la boule unité complexe n-dimensionnelle pour n2. Ensuite, on observe qu’il existe une isométrie holomorphe du produit du disque unité et de la boule unité complexe n-dimensionnelle dans tout domaine symétrique borné irréductible de rang 2 non-biholomorphe à aucun domaine de type IV. En particulier, notre étude fournit de nombreux nouveaux exemples d’isométries holomorphes du disque de Poincaré dans les domaines symétriques bornés irréductibles de rang au moins deux, à l’exception des domaines de type IV.

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DOI: 10.5802/aif.3293
Classification: 32M15, 53C55, 53C42
Keywords: Holomorphic isometries, Bounded symmetric domains
Mot clés : isométries holomorphes, domaines symétriques bornés
Chan, Shan Tai 1; Yuan, Yuan 2

1 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
2 Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, USA
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Chan, Shan Tai; Yuan, Yuan. Holomorphic isometries from the Poincaré disk into bounded symmetric domains of rank at least two. Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2205-2240. doi : 10.5802/aif.3293. http://archive.numdam.org/articles/10.5802/aif.3293/

[1] Calabi, Eugenio Isometric imbedding of complex manifolds, Ann. Math., Volume 58 (1953), pp. 1-23 | DOI | MR | Zbl

[2] Chan, Shan Tai On global rigidity of the p-th root embedding, Proc. Am. Math. Soc., Volume 144 (2016) no. 1, pp. 347-358 | DOI | MR | Zbl

[3] Chan, Shan Tai Classification problem of holomorphic isometries of the unit disk into polydisks, Mich. Math. J., Volume 66 (2017) no. 4, pp. 745-767 | DOI | MR | Zbl

[4] Chan, Shan Tai; Mok, Ngaiming Holomorphic isometries of 𝔹 m into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals, Math. Z., Volume 286 (2017) no. 1-2, pp. 679-700 | DOI | MR | Zbl

[5] Chan, Shan Tai; Xiao, Ming; Yuan, Yuan Holomorphic isometries between products of complex unit balls, Int. J. Math., Volume 28 (2017) no. 9, 1740010, 22 pages | DOI | MR | Zbl

[6] Clozel, Laurent; Ullmo, Emmanuel Correspondances modulaires et mesures invariantes, J. Reine Angew. Math., Volume 558 (2003), pp. 47-83 | DOI | MR | Zbl

[7] Ebenfelt, Peter Local holomorphic isometries of a modified projective space into a standard projective space; rational conformal factors, Math. Ann., Volume 363 (2015) no. 3-4, pp. 1333-1348 | DOI | MR | Zbl

[8] Huang, Xiaojun; Yuan, Yuan Holomorphic isometry from a Kähler manifold into a product of complex projective manifolds, Geom. Funct. Anal., Volume 24 (2014) no. 3, pp. 854-886 | DOI | MR | Zbl

[9] Hwang, Jun-Muk; To, Wing-Keung Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Am. J. Math., Volume 124 (2002) no. 6, pp. 1221-1246 | DOI | MR | Zbl

[10] Mok, Ngaiming Metric rigidity theorems on Hermitian locally symmetric manifolds, Series in Pure Mathematics, 6, World Scientific, 1989, xiv+278 pages | DOI | MR | Zbl

[11] Mok, Ngaiming Characterization of certain holomorphic geodesic cycles on quotients of bounded symmetric domains in terms of tangent subspaces, Compos. Math., Volume 132 (2002) no. 3, pp. 289-309 | DOI | MR | Zbl

[12] Mok, Ngaiming On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains, Acta Math. Sci., Ser. B, Engl. Ed., Volume 29 (2009) no. 4, pp. 881-902 | DOI | MR | Zbl

[13] Mok, Ngaiming Geometry of holomorphic isometries and related maps between bounded domains, Geometry and analysis. No. 2 (Advanced Lectures in Mathematics (ALM)), Volume 18, International Press., 2011, pp. 225-270 | MR | Zbl

[14] Mok, Ngaiming Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric, J. Eur. Math. Soc., Volume 14 (2012) no. 5, pp. 1617-1656 | DOI | MR | Zbl

[15] Mok, Ngaiming Holomorphic isometries of the complex unit ball into irreducible bounded symmetric domains, Proc. Am. Math. Soc., Volume 144 (2016) no. 10, pp. 4515-4525 | DOI | MR | Zbl

[16] Mok, Ngaiming; Ng, Sui-Chung Second fundamental forms of holomorphic isometries of the Poincaré disk into bounded symmetric domains and their boundary behavior along the unit circle, Sci. China, Ser. A, Volume 52 (2009) no. 12, pp. 2628-2646 | DOI | MR | Zbl

[17] Ng, Sui-Chung On holomorphic isometric embeddings of the unit disk into polydisks, Proc. Am. Math. Soc., Volume 138 (2010) no. 8, pp. 2907-2922 | DOI | MR | Zbl

[18] Wolf, Joseph A. Fine structure of Hermitian symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970) (Pure and Applied Mathematics), Volume 8, Marcel Dekker, 1972, pp. 271-357 | MR | Zbl

[19] Xiao, Ming; Yuan, Yuan Complexity of holomorphic maps from the complex unit ball to classical domains, Asian J. Math., Volume 22 (2018) no. 4, pp. 729-760 | DOI | MR | Zbl

[20] Xiao, Ming; Yuan, Yuan Holomorphic maps from the complex unit ball to Type IV classical domains, J. Math. Pures Appl. (2019) | DOI | Zbl

[21] Yuan, Yuan Local holomorphic isometries, old and new results, Proceedings of the Seventh International Congress of Chinese Mathematicians. Vol. II (Advanced Lectures in Mathematics (ALM)), Volume 44, International Press, 2019, pp. 409-422

[22] Yuan, Yuan; Zhang, Yuan Rigidity for local holomorphic isometric embeddings from 𝔹 n into 𝔹 N 1 ××𝔹 N m up to conformal factors, J. Differ. Geom., Volume 90 (2012) no. 2, pp. 329-349 | MR | Zbl

[23] Zhang, Fuzhen Matrix theory. Basic results and techniques, Universitext, Springer, 2011, xviii+399 pages | DOI | MR | Zbl

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