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RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bn+1 TO B3n-1
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 Title & Authors
RIGIDITY OF PROPER HOLOMORPHIC MAPS FROM Bn+1 TO B3n-1
Wang, Sung-Ho;
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 Abstract
Let be the unit ball in the complex vector space with the standard Hermitian metric. Let be the boundary sphere with the induced CR structure. Let f : be a local CR immersion. If N < 3n - 1, the asymptotic vectors of the CR second fundamental form of f at each point form a subspace of the CR(horizontal) tangent space of of codimension at most 1. We study the higher order derivatives of this relation, and we show that a linearly full local CR immersion f : , N 3n-2, can only occur when N = n, 2n, or 2n + 1. As a consequence, it gives an extension of the classification of the rational proper holomorphic maps from to by Hamada to the classification of the rational proper holomorphic maps from to .
 Keywords
CR immersion;proper holomorphic map;rigidity;
 Language
English
 Cited by
1.
Linearity and Second Fundamental Forms for Proper Holomorphic Maps from $\mathbb{B}^{n+1}$ to $\mathbb{B}^{4n-3}$, Journal of Geometric Analysis, 2012, 22, 4, 977  crossref(new windwow)
2.
Flatness of CR submanifolds in a sphere, Science China Mathematics, 2010, 53, 3, 701  crossref(new windwow)
 References
1.
R. Bryant, Bochner-Kahler metrics, J. Amer. Math. Soc. 14 (2001), no. 3, 623.715. crossref(new window)

2.
E. Cartan, Sur les varietes de courbure constante d'un espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919), 125.160; Bull. Soc. Math. France 48 (1920), 132.208

3.
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219.271 crossref(new window)

4.
J. P. D'Angelo, Proper holomorphic maps between balls of different dimensions, Michigan Math. J. 35 (1988), no. 1, 83.90 crossref(new window)

5.
J. P. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1993

6.
P. Ebenfelt, X. Huang, and D. Zaitsev, Rigidity of CR-immersions into spheres, Comm. Anal. Geom. 12 (2004), no. 3, 631.670 crossref(new window)

7.
F. Forstneric, Proper holomorphic maps from balls, Duke Math. J. 53 (1986), no. 2, 427.441 crossref(new window)

8.
H. Hamada, Rational proper holomorphic maps from Bn into B2n, Math. Ann. 331 (2005), no. 3, 693.711 crossref(new window)

9.
C.-K. Han, Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms, Math. Ann. 309 (1997), no. 3, 401.409 crossref(new window)

10.
X. Huang, On a semi-rigidity property for holomorphic maps, Asian J. Math. 7 (2003), no. 4, 463.492

11.
X. Huang and S. Ji, Mapping Bn into B2n.1, Invent. Math. 145 (2001), no. 2, 219.250 crossref(new window)

12.
X. Huang, S. Ji, and D. Xu, Several results for holomorphic mappings from Bn into BN, Geometric analysis of PDE and several complex variables, 267.292, Contemp. Math., 368, Amer. Math. Soc., Providence, RI, 2005

13.
X. Huang, S. Ji, and D. Xu, A new gap phenomenon for proper holomorphic mappings from Bn into BN, Math. Res. Lett. 13 (2006), no. 4, 515.529 crossref(new window)

14.
T. Iwatani, Kahler submanifolds with vanishing Bochner curvature tensor, Mem. Fac. Sci. Kyushu Univ. Ser. A 30 (1976), no. 2, 319.321 crossref(new window)

15.
S. H. Wang, 1-rigidity for CR submanifolds in spheres, arXiv. math.DG/0506134

16.
E. Cartan, Bull. Soc. Math. France 48 (1920), 132.208