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Patrick Ahern; Xianghong Gong
Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one
Annales de la faculté des sciences de Toulouse Sér. 6, 18 no. 1 (2009), p. 1-64, doi: 10.5802/afst.1204
Article PDF | Reviews MR 2518102 | Zbl 1182.32013
[1] Bishop (E.).— Differentiable manifolds in complex Euclidean space, Duke Math. J., 32, p. 1-22 (1965). Article | MR 200476 | Zbl 0154.08501
[2] Écalle (J.).— Les fonctions résurgentes, I, II. Publications Mathématiques d’Orsay 81, 5, 6, 1-247, p. 248-531, Université de Paris-Sud, Département de Mathématique, Orsay (1981). Article | Zbl 0499.30035
[3] Gong (X.).— On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents, Comment. Math. Helv. 69, no.4, p. 549-574 (1994). MR 1303227 | Zbl 0826.32012
[4] Gong (X.).— Real analytic submanifolds under unimodular transformations, Proc. Amer. Math. Soc. 123, no.1, p. 191-200 (1995). MR 1231299 | Zbl 0821.32012
[5] Gong (X.).— Divergence of the normalization for real Lagrangian surfaces near complex tangents, Pacific J. Math. 176, no. 2, p. 311–324 (1996). Article | MR 1434993 | Zbl 0879.32008
[6] Huang (X.) and Yin (W.).— A Bishop surface with a vanishing Bishop invariant, preprint. arXiv
[7] Malgrange (B.).— Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982, pp. 59-73, Astérisque, p. 92-93, Soc. Math. France, Paris (1982). Numdam | MR 689526 | Zbl 0526.58009
[8] Moser (J.K.) and Webster (S.M.).— Normal forms for real surfaces in ${\mathbb{C}}^2$ near complex tangents and hyperbolic surface transformations, Acta Math., 150, p. 255-296 (1983). MR 709143 | Zbl 0519.32015
[9] Voronin (S.M.).— Analytic classification of germs of conformal mappings $({\mathbb{C}},\,0)\rightarrow ({\mathbb{C}},\,0)$, Functional Anal. Appl. 15, no. 1, p. 1-13 (1981). MR 609790 | Zbl 0463.30010
[10] Voronin (S.M.).— The Darboux-Whitney theorem and related questions, in Nonlinear Stokes phenomena, p. 139–233, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, (1993). MR 1206044 | Zbl 0789.58015
[11] Webster (S.M.).— Holomorphic symplectic normalization of a real function, Ann. Scuola Norm. Sup. di Pisa, 19, p. 69-86 (1992). Numdam | MR 1183758 | Zbl 0763.58010
Patrick Ahern; Xianghong Gong
Real analytic manifolds in ${\mathbb{C}}^n$ with parabolic complex tangents along a submanifold of codimension one
Annales de la faculté des sciences de Toulouse Sér. 6, 18 no. 1 (2009), p. 1-64, doi: 10.5802/afst.1204
Article PDF | Reviews MR 2518102 | Zbl 1182.32013
Résumé - Abstract
We will classify $n$-dimensional real submanifolds in ${{\mathbb{C}}}^n$ which have a set of parabolic complex tangents of real dimension $n-1$. All such submanifolds are equivalent under formal biholomorphisms. We will show that the equivalence classes under convergent local biholomorphisms form a moduli space of infinite dimension. We will also show that there exists an $n$-dimensional submanifold $M$ in ${{\mathbb{C}}}^n$ such that its images under biholomorphisms $(z_1, \dots , z_n) \mapsto (rz_1, \dots , rz_{n-1}, r^2z_n)$, $r > 1$, are not equivalent to $M$ via any local volume-preserving holomorphic map.
Bibliography
[2] Écalle (J.).— Les fonctions résurgentes, I, II. Publications Mathématiques d’Orsay 81, 5, 6, 1-247, p. 248-531, Université de Paris-Sud, Département de Mathématique, Orsay (1981). Article | Zbl 0499.30035
[3] Gong (X.).— On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents, Comment. Math. Helv. 69, no.4, p. 549-574 (1994). MR 1303227 | Zbl 0826.32012
[4] Gong (X.).— Real analytic submanifolds under unimodular transformations, Proc. Amer. Math. Soc. 123, no.1, p. 191-200 (1995). MR 1231299 | Zbl 0821.32012
[5] Gong (X.).— Divergence of the normalization for real Lagrangian surfaces near complex tangents, Pacific J. Math. 176, no. 2, p. 311–324 (1996). Article | MR 1434993 | Zbl 0879.32008
[6] Huang (X.) and Yin (W.).— A Bishop surface with a vanishing Bishop invariant, preprint. arXiv
[7] Malgrange (B.).— Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982, pp. 59-73, Astérisque, p. 92-93, Soc. Math. France, Paris (1982). Numdam | MR 689526 | Zbl 0526.58009
[8] Moser (J.K.) and Webster (S.M.).— Normal forms for real surfaces in ${\mathbb{C}}^2$ near complex tangents and hyperbolic surface transformations, Acta Math., 150, p. 255-296 (1983). MR 709143 | Zbl 0519.32015
[9] Voronin (S.M.).— Analytic classification of germs of conformal mappings $({\mathbb{C}},\,0)\rightarrow ({\mathbb{C}},\,0)$, Functional Anal. Appl. 15, no. 1, p. 1-13 (1981). MR 609790 | Zbl 0463.30010
[10] Voronin (S.M.).— The Darboux-Whitney theorem and related questions, in Nonlinear Stokes phenomena, p. 139–233, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, (1993). MR 1206044 | Zbl 0789.58015
[11] Webster (S.M.).— Holomorphic symplectic normalization of a real function, Ann. Scuola Norm. Sup. di Pisa, 19, p. 69-86 (1992). Numdam | MR 1183758 | Zbl 0763.58010
