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

Chuaqui, Martin

doi:10.1090/S0002-9939-10-10249-4

Integral conditions on the Schwarzian for curves to be simple or unknotted
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Proc. Amer. Math. Soc. 138 (2010), 1811-1819 Request permission

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

Lars V. Ahlfors, Cross-ratios and Schwarzian derivatives in $\textbf {R}^n$, Complex analysis, Birkhäuser, Basel, 1988, pp. 1–15. MR 981397
A. D. Alexandrov and Yu. G. Reshetnyak, General theory of irregular curves, Mathematics and its Applications (Soviet Series), vol. 29, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by L. Ya. Yuzina. MR 1117220, DOI 10.1007/978-94-009-2591-5
F. Brickell and C. C. Hsiung, The total absolute curvature of closed curves in Riemannian manifolds, J. Differential Geometry 9 (1974), 177–193. MR 339032
Martin Chuaqui, On Ahlfors’ Schwarzian derivative and knots, Pacific J. Math. 231 (2007), no. 1, 51–61. MR 2304621, DOI 10.2140/pjm.2007.231.51
Martin Chuaqui and Julian Gevirtz, Simple curves in ${\Bbb R}^n$ and Ahlfors’ Schwarzian derivative, Proc. Amer. Math. Soc. 132 (2004), no. 1, 223–230. MR 2021266, DOI 10.1090/S0002-9939-03-07013-8
M. Chuaqui, P. Duren, and B. Osgood, Univalence criteria for lifts of harmonic mappings to minimal surfaces, J. Geom. Anal. 17 (2007), no. 1, 49–74. MR 2302873, DOI 10.1007/BF02922082
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.
Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490
Zeev Nehari, Conformal mapping, Dover Publications, Inc., New York, 1975. Reprinting of the 1952 edition. MR 0377031