We first study holomorphic isometries from the Poincaré disk into the product of the unit disk and the complex unit -ball for . On the other hand, we observe that there exists a holomorphic isometry from the product of the unit disk and the complex unit -ball into any irreducible bounded symmetric domain of rank which is not biholomorphic to any type- 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 except for type- domains.
Nous étudions d’abord les isométries holomorphes du disque de Poincaré dans le produit du disque unité et de la boule unité complexe -dimensionnelle pour . Ensuite, on observe qu’il existe une isométrie holomorphe du produit du disque unité et de la boule unité complexe -dimensionnelle dans tout domaine symétrique borné irréductible de rang non-biholomorphe à aucun domaine de type . 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 .
Revised:
Accepted:
Published online:
Keywords: Holomorphic isometries, Bounded symmetric domains
Mot clés : isométries holomorphes, domaines symétriques bornés
@article{AIF_2019__69_5_2205_0, author = {Chan, Shan Tai and Yuan, Yuan}, title = {Holomorphic isometries from the {Poincar\'e} disk into bounded symmetric domains of rank at least two}, journal = {Annales de l'Institut Fourier}, pages = {2205--2240}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {69}, number = {5}, year = {2019}, doi = {10.5802/aif.3293}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.3293/} }
TY - JOUR AU - Chan, Shan Tai AU - Yuan, Yuan TI - Holomorphic isometries from the Poincaré disk into bounded symmetric domains of rank at least two JO - Annales de l'Institut Fourier PY - 2019 SP - 2205 EP - 2240 VL - 69 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.3293/ DO - 10.5802/aif.3293 LA - en ID - AIF_2019__69_5_2205_0 ER -
%0 Journal Article %A Chan, Shan Tai %A Yuan, Yuan %T Holomorphic isometries from the Poincaré disk into bounded symmetric domains of rank at least two %J Annales de l'Institut Fourier %D 2019 %P 2205-2240 %V 69 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.3293/ %R 10.5802/aif.3293 %G en %F AIF_2019__69_5_2205_0
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] Isometric imbedding of complex manifolds, Ann. Math., Volume 58 (1953), pp. 1-23 | DOI | MR | Zbl
[2] On global rigidity of the -th root embedding, Proc. Am. Math. Soc., Volume 144 (2016) no. 1, pp. 347-358 | DOI | MR | Zbl
[3] 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] Holomorphic isometries of 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] Holomorphic isometries between products of complex unit balls, Int. J. Math., Volume 28 (2017) no. 9, 1740010, 22 pages | DOI | MR | Zbl
[6] Correspondances modulaires et mesures invariantes, J. Reine Angew. Math., Volume 558 (2003), pp. 47-83 | DOI | MR | Zbl
[7] 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] 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] Volumes of complex analytic subvarieties of Hermitian symmetric spaces, Am. J. Math., Volume 124 (2002) no. 6, pp. 1221-1246 | DOI | MR | Zbl
[10] Metric rigidity theorems on Hermitian locally symmetric manifolds, Series in Pure Mathematics, 6, World Scientific, 1989, xiv+278 pages | DOI | MR | Zbl
[11] 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] 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] 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] 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] 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] 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] 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] 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] 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] Holomorphic maps from the complex unit ball to Type IV classical domains, J. Math. Pures Appl. (2019) | DOI | Zbl
[21] 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] Rigidity for local holomorphic isometric embeddings from into up to conformal factors, J. Differ. Geom., Volume 90 (2012) no. 2, pp. 329-349 | MR | Zbl
[23] Matrix theory. Basic results and techniques, Universitext, Springer, 2011, xviii+399 pages | DOI | MR | Zbl
Cited by Sources: