Arkiv för Matematik

  • Ark. Mat.
  • Volume 43, Number 2 (2005), 347-364.

Minimizing singularities of generic plane disks with immersed boundaries

Tobias Ekholm and Ola Larsson

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Abstract

A cooriented circle immersion into the plane can be extended to a stable map of the disk which is an immersion in a neighborhood of the boundary and with outward normal vector field along the boundary equal to the given coorienting normal vector field. We express the minimal number of fold components of such a stable map as a function of its number of cusps and of the normal degree of its boundary. We also show that this minimum is attained for any cooriented circle immersion of normal degree not equal to one.

Note

The first author is a research fellow of the Royal Swedish Academy of Sciences sponsored by the Knut and Alice Wallenberg foundation.

Article information

Source
Ark. Mat., Volume 43, Number 2 (2005), 347-364.

Dates
Received: 8 January 2004
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898904

Digital Object Identifier
doi:10.1007/BF02384784

Mathematical Reviews number (MathSciNet)
MR2173956

Zentralblatt MATH identifier
1101.58024

Rights
2005 © Institut Mittag-Leffler

Citation

Ekholm, Tobias; Larsson, Ola. Minimizing singularities of generic plane disks with immersed boundaries. Ark. Mat. 43 (2005), no. 2, 347--364. doi:10.1007/BF02384784. https://projecteuclid.org/euclid.afm/1485898904


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References

  • Arnold, V. I., Gusein-Zade, S. M. and Varchenko, A. N., Singularities of Differentiable Maps, Vol. 1, Monogr. Math. 82, Birkhäuser, Boston, MA, 1985.
  • Eliashberg, Ya., On singularities of folding type. Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 1110–1126 (Russian). English transl.: Math. USSR-Izv. 4 (1970), 1119–1134.
  • Haefliger, A., Quelques remarques sur les applications différentiables d'une surface dans le plan, Ann. Inst. Fourier. Grenoble 10 (1960), 47–60.
  • Poénaru, V., Extensions des immersions en codimension 1 (d'apres Samuel Blanck), Sem. Bourbaki 10., Exp. 342, Benjamin, New York-Amsterdam, 1969.
  • Whitney, H., On regular closed curves in the plane, Compositio Math. 4 (1936), 276–284.