Rigidity of proper holomorphic maps from B^(n+1) to B^(3n-1)
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Let B^{n+1} be the unit ball in the complex vector space C^{n+1} with the standard Hermitian metric. Let ∑^n = ∂B^{n+1} = S^{2n+1} be the boundary sphere with the induced CR structure. Let f : ∑^n→∑^{N} 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 ∑^n 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→∑^{N}, 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 B^{n+1} to B^{2n+2} by Hamada to the classification of the rational proper holomorphic maps from B^{n+1} to B^{3n-1}.
Abstract
Let B^{n+1} be the unit ball in the complex vector space C^{n+1} with the standard Hermitian metric. Let ∑^n = ∂B^{n+1} = S^{2n+1} be the boundary sphere with the induced CR structure. Let f : ∑^n→∑^{N} 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 ∑^n 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→∑^{N}, 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 B^{n+1} to B^{2n+2} by Hamada to the classification of the rational proper holomorphic maps from B^{n+1} to B^{3n-1}.
참고문헌 (15)
- R. Bryant, "Bochner-K¨ahler metrics", J. Amer. Math. Soc., 14(3), pp.623~715 (2001)
- E. Cartan, "Sur les vari´et´es de courbure constante d’un espace euclidien ou non-euclidien", Bull. Soc. Math. France, 48, pp.132~208 (1919)
- S. S. Chern, "Real hypersurfaces in complex manifolds", Acta Math., 133, pp.219~271 (1974)
- J. P. D’Angelo, "Proper holomorphic maps between balls of different dimensions", Michigan Math. J., 35(1), pp.83~90 (1988)
- J. P. D’Angelo, "Several Complex Variables and the Geometry of Real Hypersurfaces, Studies in Advanced Mathematics" (1993), CRC Press
- P. Ebenfelt, "Rigidity of CR-immersions into spheres", Comm. Anal. Geom., 12(3), pp.631~670 (2004)
- F. Forstneriˇc, "Proper holomorphic maps from balls", Duke Math. J., 53(2), pp.427~441 (1986)
- H. Hamada, "ational proper holomorphic maps from Bn into B2n", Math. Ann., 331(3), pp.693~711 (2005)
- C.-K. Han, "Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms", Math. Ann., 309(3), pp.401~409 (1997)
- X. Huang, "On a semi-rigidity property for holomorphic maps", Asian J. Math., 7(4), pp.463~492 (2003)
- X. Huang, "Invent. Math", 219–250., 145(2), pp.219~250 (2001)
- X. Huang, "Several results for holomorphic mappings from Bn into BN, Geometric analysis of PDE and several complex variables", Amer. Math. Soc., 368, pp.267~292 (2005)
- X. Huang, "A new gap phenomenon for proper holomorphic mappings from Bn into BN", Math. Res. Lett., 13(4), pp.515~529 (2006)
- T. Iwatani, "K¨ahler submanifolds with vanishing Bochner curvature tensor", Mem. Fac. Sci. Kyushu Univ. Ser. A, 30(2), pp.319~321 (1976)
- S. H. Wang, "1-rigidity for CR submanifolds in spheres", ~