## Kyoto Journal of Mathematics

### Infinitesimal CR automorphisms and stability groups of infinite-type models in $\mathbb{C}^{2}$

#### Abstract

The purpose of this article is to give explicit descriptions for stability groups of real rigid hypersurfaces of infinite type in $\mathbb{C}^{2}$. The decompositions of infinitesimal CR automorphisms are also given.

#### Article information

Source
Kyoto J. Math., Volume 56, Number 2 (2016), 441-464.

Dates
Revised: 30 March 2015
Accepted: 15 April 2015
First available in Project Euclid: 10 May 2016

https://projecteuclid.org/euclid.kjm/1462901084

Digital Object Identifier
doi:10.1215/21562261-3478925

Mathematical Reviews number (MathSciNet)
MR3500847

Zentralblatt MATH identifier
1351.32060

#### Citation

Hayashimoto, Atsushi; Thu, Ninh Van. Infinitesimal CR automorphisms and stability groups of infinite-type models in $\mathbb{C}^{2}$. Kyoto J. Math. 56 (2016), no. 2, 441--464. doi:10.1215/21562261-3478925. https://projecteuclid.org/euclid.kjm/1462901084

#### References

• [A] M. Abate, “Discrete holomorphic local dynamical systems” in Holomorphic Dynamical Systems, Lecture Notes in Math. 1998, Springer, Berlin, 2010, 1–55.
• [B] F. Bracci, Local dynamics of holomorphic diffeomorphisms, Boll. Unione Mat. Ital. 7 (2004), 609–636.
• [CM] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271.
• [D] J. P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), 615–637.
• [EKS1] V. Ezhov, M. Kolář, and G. Schmalz, Degenerate hypersurfaces with a two-parametric family of automorphisms, Complex Var. Elliptic Equ. 54 (2009), 283–291.
• [EKS2] V. Ezhov, M. Kolář, and G. Schmalz, Normal forms and symmetries of real hypersurfaces of finite type in $\mathbb{C}^{2}$, Indiana Univ. Math. J. 62 (2013), 1–32.
• [KN] K.-T. Kim and N. V. Thu, On the tangential holomorphic vector fields vanishing at an infinite type point, Trans. Amer. Math. Soc. 367 (2015), 867–885.
• [K1] M. Kolář, Normal forms for hypersurfaces of finite type in $\mathbb{C}^{2}$, Math. Res. Lett. 12 (2005), 897–910.
• [K2] M. Kolář, Local symmetries of finite type hypersurfaces in $\mathbb{C}^{2}$, Sci. China Ser. A 49 (2006), 1633–1641.
• [K3] M. Kolář, Local equivalence of symmetric hypersurfaces in $\mathbb{C}^{2}$, Trans. Amer. Math. Soc. 362, no. 6 (2010), 2833–2843.
• [KM] M. Kolář and F. Meylan, Infinitesimal CR automorphisms of hypersurfaces of finite type in $\mathbb{C}^{2}$, Arch. Math. (Brno) 47 (2011), 367–375.
• [KMZ] M. Kolář, F. Meylan, and D. Zaitsev, Chern-Moser operators and polynomial models in CR geometry, Adv. Math. 263 (2014), 321–356.
• [N1] V. T. Ninh, On the existence of tangential holomorphic vector fields vanishing at an infinite type point, preprint, arXiv:1303.6156v7 [math.CV].
• [N2] V. T. Ninh, On the CR automorphism group of a certain hypersurface of infinite type in $\mathbb{C}^{2}$, Complex Var. Elliptic Equ. 60 (2015), 977–991.
• [NCM] V. T. Ninh, V. T. Chu, and A. D. Mai, On the real-analytic infinitesimal CR automorphism of hypersurfaces of infinite type, preprint, arXiv:1404.4914v2 [math.CV].
• [S1] N. K. Stanton, Infinitesimal CR automorphisms of rigid hypersurfaces, Amer. J. Math. 117 (1995), 141–167.
• [S2] N. K. Stanton, Infinitesimal CR automorphisms of real hypersurfaces, Amer. J. Math. 118 (1996), 209–233.