Infinitesimal CR automorphisms for a class of polynomial models

Kolář, Martin   ; Meylan, Francine

Archivum mathematicum, Tome 53 (2017) no. 5, pp. 255-265

Abstract (VO)

In this paper we study infinitesimal CR automorphisms of Levi degenerate hypersurfaces. We illustrate the recent general results of [18], [17], [15], on a class of concrete examples, polynomial models in $\mathbb{C}^3$ of the form $\Im \; w = \Re (P(z) \overline{Q(z)}) $, where $P$ and $Q$ are weighted homogeneous holomorphic polynomials in $z = (z_1, z_2)$. We classify such models according to their Lie algebra of infinitesimal CR automorphisms. We also give the first example of a non monomial model which admits a nonlinear rigid automorphism.

Détail

DOI : 10.5817/AM2017-5-255

Classification : 32V35, 32V40
Keywords: Levi degenerate hypersurfaces; finite multitype; polynomial models; infinitesimal CR automorphisms

Bibliographie

[1] Baouendi, M.S., Ebenfelt, P., Rothschild, L.P.: Local geometric properties of real submanifolds in complex space. Bull. Amer. Math. Soc. (N.S.) 37 (3) (2000), 309–336. | DOI | MR

[2] Bedford, E., Pinchuk, S.I.: Convex domains with noncompact groups of automorphisms. Mat. Sb. 185 (1994), 3–26. | MR

[3] Bloom, T., Graham, I.: On “type” conditions for generic real submanifolds of $C^{n}$. Invent. Math. 40 (3) (1977), 217–243. | DOI | MR

[4] Cartan, E.: Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I. Ann. Math. Pura Appl. 11 (1932), 17–90. | DOI | MR

[5] Cartan, E.: Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, II. Ann. Scuola Norm. Sup. Pisa 1 (1932), 333–354. | MR

[6] Catlin, D.: Boundary invariants of pseudoconvex domains. Ann. of Math. (2) 120 (1984), 529–586. | MR | Zbl

[7] Chern, S. S., Moser, J.: Real hypersurfaces in complex manifolds. Acta Math. 133 (1974), 219–271. | DOI | MR

[8] D’Angelo, J.: Orders od contact, real hypersurfaces and applications. Ann. of Math. (2) 115 (1982), 615–637. | MR

[9] Kim, S.Y., Zaitsev, D.: Equivalence and embedding problems for CR-structures of any codimension. Topology 44 (2005), 557–584. | DOI | MR | Zbl

[10] Kohn, J.J.: Boundary behaviour of $\bar{\partial }$ on weakly pseudoconvex manifolds of dimension two. J. Differential Geom. 6 (1972), 523–542. | DOI | MR

[11] Kolář, M.: Normal forms for hypersurfaces of finite type in $\mathbb{C}^2$. Math. Res. Lett. 12 (2005), 523–542. | DOI | MR

[12] Kolář, M.: The Catlin multitype and biholomorphic equivalence of models. Internat. Math. Res. Notices 18 (2010), 3530–3548. | DOI | MR | Zbl

[13] Kolář, M., Kossovskiy, I., Zaitsev, D.: Normal forms in Cauchy-Riemann geometry. Analysis and geometry in several complex variables, vol. 681, Contemp. Math., 2017, pp. 153–177. | MR | Zbl

[14] Kolář, M., Lamel, B.: Ruled hypersurfaces in $\mathbb{C}^{2}$. J. Geom. Anal. 25 (2015), 1240–1281. | DOI | MR | Zbl

[15] Kolář, M., Meylan, F.: Nonlinear CR automorphisms of Levi degenerate hypersurfaces and a new gap phenomenon. arXiv : 1703.07123 [CV].

[16] Kolář, M., Meylan, F.: Chern-Moser operators and weighted jet determination problems. Geometric analysis of several complex variables and related topics, vol. 550, Contemp. Math., 2011, pp. 75–88. | MR | Zbl

[17] Kolář, M., Meylan, F.: Higher order symmetries of real hypersurfaces in $\mathbb{C}^3$. Proc. Amer. Math. Soc. 144 (2016), 4807–4818. | DOI | MR | Zbl

[18] Kolář, M., Meylan, F., Zaitsev, D.: Chern-Moser operators and polynomial models in CR geometry. Adv. Math. 263 (2014), 321–356. | DOI | MR | Zbl

[19] Poincaré, H.: Les fonctions analytique de deux variables et la représentation conforme. Rend. Circ. Mat. Palermo 23 (1907), 185–220. | DOI

[20] Vitushkin, A.G.: Real analytic hypersurfaces in complex manifolds. Russian Math. Surveys 40 (1985), 1–35. | DOI | MR | Zbl

[21] Webster, S.M.: On the Moser normal form at a non-umbilic point. Math. Ann. 233 (1978), 97–102. | DOI | MR | Zbl