Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Integral conditions on the Schwarzian for curves to be simple or unknotted


Author: Martin Chuaqui
Journal: Proc. Amer. Math. Soc. 138 (2010), 1811-1819
MSC (2000): Primary 53A04, 53A55; Secondary 34C10
DOI: https://doi.org/10.1090/S0002-9939-10-10249-4
Published electronically: January 6, 2010
MathSciNet review: 2587466
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: By considering integral bounds on the Schwarzian derivative we extend previous results on sufficient conditions for curves in euclidean spaces to be simple or unknotted. The conditions are optimal.


References [Enhancements On Off] (What's this?)

  • [Ah] L.V. Ahlfors, Cross-ratios and Schwarzian derivatives in $ \mathbb{R}^n$, Complex Analysis, 1-15, Birkhäuser, Basel, 1988. MR 981397 (90a:30055)
  • [AR] A.D. Alexandrov and Yu.G. Reshetnyak, General theory of irregular curves, Kluwer Academic, Dordrecht, 1989. MR 1117220 (92h:53003)
  • [BH] F. Brickell and C.C. Hsiung, The total curvature of closed curves in Riemannian manifolds, J. Diff. Geom. 9 (1974), 177-193. MR 0339032 (49:3795)
  • [Ch] M. Chuaqui, On Ahlfors' Schwarzian derivative and knots, Pacific J. Math. 231 (2007), 51-62. MR 2304621 (2008m:53003)
  • [ChG] M. Chuaqui and J. Gevirtz, Simple curves in $ \mathbb{R}^n$ and Ahlfors' Schwarzian derivative, Proc. Amer. Math. Soc. 132 (2004), 223-230. MR 2021266 (2005f:53001)
  • [ChDO1] M. Chuaqui, P. Duren and B. Osgood, Univalence criteria for lifts of harmonic mappings to minimal surfaces, J. Geometric Analysis 17 (2007), 49-74. MR 2302873 (2008d:31001)
  • [ChDO2] M. Chuaqui, P. Duren and B. Osgood, Injectivity criteria for holomorphic curves in $ \mathbb{C}^n$, Pure and Applied Mathematics Quarterly 7 (2011), no. 1, 215-243.
  • [H] P. Hartman, Ordinary differential equations, $ 2^{\mbox{\tiny nd}}$ Edition, Birkhäuser, Boston, MA, 1982. MR 658490 (83e:34002)
  • [N] Z. Nehari, Conformal mapping, Dover Books on Advanced Mathematics, Dover Publ. Inc., New York, 1975. MR 0377031 (51:13206)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53A04, 53A55, 34C10

Retrieve articles in all journals with MSC (2000): 53A04, 53A55, 34C10


Additional Information

Martin Chuaqui
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna 4860, Santiago, Chile
Email: mchuaqui@mat.puc.cl

DOI: https://doi.org/10.1090/S0002-9939-10-10249-4
Keywords: Ahlfors' Schwarzian, simple curves, knots, M\"{o}bius transformation
Received by editor(s): November 24, 2008
Published electronically: January 6, 2010
Additional Notes: This work was partially supported by Fondecyt Grant 1071019.
Communicated by: Mario Bank
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society