| Title:
|
The local equivalence problem in CR geometry (English) |
| Author:
|
Kolář, Martin |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
42 |
| Issue:
|
5 |
| Year:
|
2006 |
| Pages:
|
253-266 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This article is dedicated to the centenary of the local CR equivalence problem, formulated by Henri Poincaré in 1907. The first part gives an account of Poincaré’s heuristic counting arguments, suggesting existence of infinitely many local CR invariants. Then we sketch the beautiful completion of Poincaré’s approach to the problem in the work of Chern and Moser on Levi nondegenerate hypersurfaces. The last part is an overview of recent progress in solving the problem on Levi degenerate manifolds. (English) |
| MSC:
|
32V40 |
| idZBL:
|
Zbl 1164.32307 |
| idMR:
|
MR2322412 |
| . |
| Date available:
|
2008-06-06T22:49:40Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108032 |
| . |
| Reference:
|
[BER] Baouendi M. S., Ebenfelt P., Rothschild L. P.: Convergence and finite determination of formal CR mappings.J. Amer. Math. Soc. 13 (2000), 697–723. Zbl 0958.32033, MR 1775734 |
| Reference:
|
[BER2] 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. MR 1754643 |
| Reference:
|
[BB] Barletta E., Bedford E.: Existence of proper mappings from domains in $\mathbb{C}^2$.Indiana Univ. Math. J. 2 (1990), 315–338. MR 1089041 |
| Reference:
|
[BFG] Beals M., Fefferman C., Grossman R.: Strictly pseudoconvex domains in $\mathbb{C}^n$.Bull. Amer. Math. Soc. 8 (1983), 125–322. MR 0684898 |
| Reference:
|
[B] Beloshapka V. K.: On the dimension of the group of automorphisms of an analytic hypersurface.Math. USSR, Izv. 14 (1980), 223–245. Zbl 0456.32015, MR 0534592 |
| Reference:
|
[BE] Beloshapka V. K., Ezhov V. V.: Normal forms and model hypersurfaces in $\mathbb{C}^2$.preprint. |
| Reference:
|
[C1] Cartan E.: Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, I.Ann. Math. Pura Appl. 11 (1932), 17–90. |
| Reference:
|
[C2] Cartan E.: Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes, II.Ann. Scuola Norm. Sup. Pisa 1 (1932), 333–354. MR 1556687 |
| Reference:
|
[CM] Chern S. S., Moser J.: Real hypersurfaces in complex manifolds.Acta Math. 133 (1974), 219–271. MR 0425155 |
| Reference:
|
[D] D’Angelo J. P.: Orders od contact, real hypersurfaces and applications.Ann. of Math. (2) 115 (1982), 615–637. MR 0657241 |
| Reference:
|
[E] Ebenfelt P.: New invariant tensors in CR structures and a normal form for real hypersurfaces at a generic Levi degeneracy.J. Differential Geom. 50 (1998), 207–247. Zbl 0945.32020, MR 1684982 |
| Reference:
|
[ELZ] Ebenfelt P., Lamel B., Zaitsev D.: Degenerate real hypersurfaces in $\mathbb{C}2$ with few automorphisms.arXiv:math.CV/0605540. |
| Reference:
|
[EHZ] Ebenfelt P., Huang X., Zaitsev D.: The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics.Amer. J. Math. 127 (2005), 169–191. MR 2115664 |
| Reference:
|
[F] Fefferman C.: Parabolic invariant theory in complex analysis.Adv. Math. 31 (1979), 131–262. Zbl 0444.32013, MR 0526424 |
| Reference:
|
[IK] Isaev A. V., Krantz S. G.: Domains with non-compact automorphism group: a survey.Adv. Math. 146 (1) (1999), 1–38. Zbl 1040.32019, MR 1706680 |
| Reference:
|
[J] Jacobowitz H.: An introduction to CR structures.Math. Surveys Monogr. 32, AMS 1990. Zbl 0712.32001, MR 1067341 |
| Reference:
|
[Ju] Juhlin R.: .PhD-thesis, UCSD Zbl 1219.32020 |
| Reference:
|
[K] Kohn J. J.: Boundary behaviour of $\bar{\partial }$ on weakly pseudoconvex manifolds of dimension two.J. Differential Geom. 6 (1972), 523–542. MR 0322365 |
| Reference:
|
[Ko1] Kolář M.: Normal forms for hypersurfaces of finite type in $ \mathbb{C}^2$.Math. Res. Lett. 12 (2005), 523–542. MR 2189248 |
| Reference:
|
[Ko2] Kolář M.: Local symmetries of finite type hypersurfaces in $\mathbb{C}^2$.Sci. China A 48 (2006), 1633–1641. MR 2288220 |
| Reference:
|
[Kow] Kowalski R.: A hypersurface in $\mathbb{C}^2$ whose stability group is not determined by 2-jets.Proc. Amer. Math. Soc. 130 (12) (2002), 3679–3686. (electronic) MR 1920048 |
| Reference:
|
[KL] Kruzhilin N. G., Loboda A. V.: Linearization of local automorphisms of pseudoconvex surfaces.Dokl. Akad. Nauk SSSR 271 (1983), 280–282. Zbl 0582.32040, MR 0718188 |
| Reference:
|
[Po] Poincaré H.: Les fonctions analytique de deux variables et la représentation conforme.Rend. Circ. Mat. Palermo 23 (1907), 185–220. |
| Reference:
|
[S] Segre B.: Intorno al problem di Poincaré della rappresentazione pseudo-conform.Rend. Accad. Lincei 13 (1931), 676–683. |
| Reference:
|
[St] Stanton N.: A normal form for rigid hypersurfaces in $\mathbb{C}^2$.Amer. J. Math. 113 (1991), 877–910. MR 1129296 |
| Reference:
|
[V] Vitushkin A. G.: Real analytic hypersurfaces in complex manifolds.Russ. Math. Surv. 40 (1985), 1–35. Zbl 0588.32025, MR 0786085 |
| Reference:
|
[W] Webster S. M.: On the Moser normal form at a non-umbilic point.Math. Ann. 233 (1978), 97–102. Zbl 0358.32013, MR 0486511 |
| Reference:
|
[We] Wells R. O., Jr.: The Cauchy-Riemann equations and differential geometry.Bull. Amer. Math. Soc. (N.S.) 6 (2) (1982), 187–199. Zbl 0496.32012, MR 0640945 |
| Reference:
|
[Wo] Wong P.: A construction of normal forms for weakly pseudoconvex CR manifolds in $\mathbb{C}^2$.Invent. Math. 69 (1982), 311–329. MR 0674409 |
| Reference:
|
[Z] Zaitsev D.: Unique determination of local CR-maps by their jets: A survey.Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Suppl. 13 (2002), 295–305. Zbl 1098.32018, MR 1984108 |
| . |
