바로가기메뉴

본문 바로가기 주메뉴 바로가기

logo

ISSN : 1226-0657

Article Contents

Prev Next
e-Submission

Vol.22 No.1

Citation Share

SPHERES IN THE SHILOV BOUNDARIES OF BOUNDED SYMMETRIC DOMAINS

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2015, v.22 no.1, pp.35-56
https://doi.org/10.7468/jksmeb.2015.22.1.35
Kim, Sung-Yeon
Kim,, S. (2015). SPHERES IN THE SHILOV BOUNDARIES OF BOUNDED SYMMETRIC DOMAINS. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 22(1), 35-56, https://doi.org/10.7468/jksmeb.2015.22.1.35

Abstract

In this paper, we classify all nonconstant smooth CR maps from a sphere <TEX>$S_{n,1}{\subset}\mathbb{C}^n$</TEX> with n > 3 to the Shilov boundary <TEX>$S_{p,q}{\subset}\mathbb{C}^{p{\times}q}$</TEX> of a bounded symmetric domain of Cartan type I under the condition that p - q < 3n - 4. We show that they are either linear maps up to automorphisms of <TEX>$S_{n,1}$</TEX> and <TEX>$S_{p,q}$</TEX> or D'Angelo maps. This is the first classification of CR maps into the Shilov boundary of bounded symmetric domains other than sphere that includes nonlinear maps.

keywords
bonded symmetric domains, Shilov boundary, CR embedding, totally geodesic embedding, Whitney map

Reference

1.

Huang, X.;Ji, S.;. (2001). Mapping Bn into B2n&#x2212;1. Invent. Math., 145(2), 219-250. 10.1007/s002220100140.

2.

Huang, X.;Ji, S.;Xu, D.;. (2006). A new gap phenomenon for proper holomorphic mappings from Bn into BN. Math. Res. Lett., 13(4), 515-529. 10.4310/MRL.2006.v13.n4.a2.

3.

Poincar&#xE9;, H.;. (1907). Les fonctions analytiques de deux variables et la representation conforme. Rend. Circ. Mat. Palermo, 23(2), 185-220. 10.1007/BF03013518.

4.

Kaup, W.;Zaitsev, D.;. (2000). On symmetric Cauchy-Riemann manifolds. Adv. Math., 149(2), 145-181. 10.1006/aima.1999.1863.

5.

Kim, S.;Zaitsev, D.;. (2013). Rigidity of CR maps between Shilov boundaries of bounded symmetric domains. Invent. Math., 193(2), 409-437. 10.1007/s00222-012-0430-3.

6.

Mok, N.;. Series in Pure Math.;Metric Rigidity Theorems on Hermitian Locally Symmetric Spaces.

7.

Seo, Aeryeong;. New examples of proper holomorphic maps between bounded symmetric domains.

8.

Tsai, I-H.;. (1993). Rigidity of proper holomorphic maps between symmetric domains. J. Differential Geom., 37(1), 123-160.

9.

Alexander, H.;. (1974). Holomorphic mappings from the ball and polydisc. Math. Ann., 209, 249-256. 10.1007/BF01351851.

10.

Chern, S.S;Moser, J.K.;. (1974). Real hypersurfaces in complex manifolds. Acta Math., 133, 219-271. 10.1007/BF02392146.

11.

Ebenfelt, P.;Huang, X.;Zaitsev, D.;. (2004). Rigidity of CR-immersions into spheres. Comm. in Analysis and Geometry, 12(3), 631-670. 10.4310/CAG.2004.v12.n3.a6.

12.

Forstneri&#x10D;, F.;. (1986). Proper holomorphic maps between balls. Duke Math. J., 53, 427-440. 10.1215/S0012-7094-86-05326-3.

13.

Huang, X.;. (2003). On a semi-rigidity property for holomorphic maps. Asian J. Math., 7(4), 463-492.

14.

Webster, S.M.;. (1979). The rigidity of C-R hypersurfaces in a sphere. Indiana Univ. Math. J., 28, 405-416. 10.1512/iumj.1979.28.28027.

15.

Faran V, J.J.;. (1986). On the linearity of proper maps between balls in the lower codimensional case. J. Differential Geom., 24, 15-17.

16.

Huang, X.;. (1999). On a linearity problem for proper holomorphic maps between balls in complex spaces of different dimensions. J. Differential Geom., 51, 13-33.

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

e-Submission OA Policy