Geodesic
On local geometry of finite multitype hypersurfaces
Archivum mathematicum, Tome 43 (2007) no. 5, pp. 459-466 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This paper studies local geometry of hypersurfaces of finite multitype. Catlin’s definition of multitype is applied to a general smooth hypersurface in $\mathbb C^{n+1}$. We prove biholomorphic equivalence of models in dimension three and describe all biholomorphisms between such models. A finite constructive algorithm for computing multitype is described. Analogous results for decoupled hypersurfaces are given.
This paper studies local geometry of hypersurfaces of finite multitype. Catlin’s definition of multitype is applied to a general smooth hypersurface in $\mathbb C^{n+1}$. We prove biholomorphic equivalence of models in dimension three and describe all biholomorphisms between such models. A finite constructive algorithm for computing multitype is described. Analogous results for decoupled hypersurfaces are given.
Classification : 32V15, 32V35, 32V40, 32Vxx
Keywords: finite type; Catlin’s multitype; model hypersurfaces; biholomorphic equivalence; decoupled domains 
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Kolář, Martin. On local geometry of finite multitype hypersurfaces. Archivum mathematicum, Tome 43 (2007) no. 5, pp. 459-466. http://geodesic.mathdoc.fr/item/ARM_2007_43_5_a9/

[1] Bloom T., Graham I.: A geometric characterization of points of type $m$ on real submanifolds of $C^{n}$. J. Differential Geometry 12 (1977), no. 2, 171–182. | MR

[2] Bloom T.: On the contact between complex manifolds and real hyp in $C^{3}$. Trans. Amer. Math. Soc. 263 (1981), no. 2, 515–529. | MR

[3] Boas H. P., Straube E. J., Yu J. Y.: Boundary limits of the Bergman kernel and metric. Michigan Math. J. 42 (1995), no. 3, 449–461. | MR | Zbl

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

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

[6] Diedrich K., Herbort G.: Pseudoconvex domains of semiregular type. in Contributions to Complex Analysis and Analytic geometry (1994), 127–161. | MR

[7] Diedrich K., Herbort G.: An alternative proof of a theorem by Boas-Straube-Yu. in Complex Analysis and Geometry, Trento 1995, Pitman Research Notes Math. Ser.

[8] Fornaess J. E., Stensones B.: Lectures on Counterexamples in Several Complex Variables. Princeton Univ. Press 1987. | MR

[9] Isaev A., Krantz S. G.: Domains with non-compact automorphism groups: a survey. Adv. Math. 146 (1999), 1–38. | MR

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

[11] Kolář M.: Convexifiability and supporting functions in ${\mathbb{C}}^2$. Math. Res. Lett. 2 (1995), 505–513. | MR

[12] Kolář M.: Generalized models and local invariants of Kohn Nirenberg domains. to appear in Math. Z. | MR | Zbl

[13] Kolář M.: On local convexifiability of type four domains in ${\mathbb{C}}^2$. Differential Geometry and Applications, Proceeding of Satellite Conference of ICM in Berlin 1999, 361–371. | MR

[14] Kolář M.: Necessary conditions for local convexifiability of pseudoconvex domains in ${\mathbb{C}}^2$. Rend. Circ. Mat. Palermo 69 (2002), 109–116. | MR

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

[16] Nikolov N.: Biholomorphy of the model domains at a semiregular boundary point. C.R. Acad. Bulgare Sci. 55 (2002), no. 5, 5–8. | MR | Zbl

[17] Yu J.: Peak functions on weakly pseudoconvex domains. Indiana Univ. Math. J. 43 (1994), no. 4, 1271–1295. | MR | Zbl