On local geometry of finite multitype hypersurfaces

Martin Kolář

Archivum Mathematicum (2007)

  • Volume: 043, Issue: 5, page 459-466
  • ISSN: 0044-8753

Abstract

top
This paper studies local geometry of hypersurfaces of finite multitype. Catlin’s definition of multitype is applied to a general smooth hypersurface in 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.

How to cite

top

Kolář, Martin. "On local geometry of finite multitype hypersurfaces." Archivum Mathematicum 043.5 (2007): 459-466. <http://eudml.org/doc/250169>.

@article{Kolář2007,
abstract = {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.},
author = {Kolář, Martin},
journal = {Archivum Mathematicum},
keywords = {finite type; Catlin’s multitype; model hypersurfaces; biholomorphic equivalence; decoupled domains; finite type; Catlin's multitype; model hypersurfaces; biholomorphic equivalence; decoupled domains},
language = {eng},
number = {5},
pages = {459-466},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On local geometry of finite multitype hypersurfaces},
url = {http://eudml.org/doc/250169},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Kolář, Martin
TI - On local geometry of finite multitype hypersurfaces
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 5
SP - 459
EP - 466
AB - 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.
LA - eng
KW - finite type; Catlin’s multitype; model hypersurfaces; biholomorphic equivalence; decoupled domains; finite type; Catlin's multitype; model hypersurfaces; biholomorphic equivalence; decoupled domains
UR - http://eudml.org/doc/250169
ER -

References

top
  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. (1977) MR0492369
  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. (1981) MR0594423
  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. (1995) Zbl0853.32028MR1357618
  4. Catlin D., Boundary invariants of pseudoconvex domains, Ann. Math. 120 (1984), 529–586. (1984) Zbl0583.32048MR0769163
  5. D’Angelo J., Orders od contact, real hypersurfaces and applications, Ann. Math. 115 (1982), 615–637. (1982) MR0657241
  6. Diedrich K., Herbort G., Pseudoconvex domains of semiregular type, in Contributions to Complex Analysis and Analytic geometry (1994), 127–161. (1994) MR1319347
  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. (1995) 
  8. Fornaess J. E., Stensones B., Lectures on Counterexamples in Several Complex Variables, Princeton Univ. Press 1987. (1987) MR0895821
  9. Isaev A., Krantz S. G., Domains with non-compact automorphism groups: a survey, Adv. Math. 146 (1999), 1–38. (1999) MR1706680
  10. Kohn J. J., Boundary behaviour of ¯ on weakly pseudoconvex manifolds of dimension two, J. Differential Geom. 6 (1972), 523–542. (1972) MR0322365
  11. Kolář M., Convexifiability and supporting functions in 2 , Math. Res. Lett. 2 (1995), 505–513. (1995) MR1355711
  12. Kolář M., Generalized models and local invariants of Kohn Nirenberg domains, to appear in Math. Z. Zbl1137.32014MR2390081
  13. Kolář M., On local convexifiability of type four domains in 2 , Differential Geometry and Applications, Proceeding of Satellite Conference of ICM in Berlin 1999, 361–371. (1999) MR1708924
  14. Kolář M., Necessary conditions for local convexifiability of pseudoconvex domains in 2 , Rend. Circ. Mat. Palermo 69 (2002), 109–116. MR1972429
  15. Kolář M., Normal forms for hypersurfaces of finite type in 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. Zbl1010.32018MR1938822
  17. Yu J., Peak functions on weakly pseudoconvex domains, Indiana Univ. Math. J. 43 (1994), no. 4, 1271–1295. (1994) Zbl0828.32003MR1322619

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.