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Simple curves in $\mathbb{R} ^n$ and Ahlfors' Schwarzian derivative

Author(s): Martin Chuaqui; Julian Gevirtz
Journal: Proc. Amer. Math. Soc. 132 (2004), 223-230.
MSC (2000): Primary 53A04, 53A55; Secondary 30C55
Posted: June 12, 2003
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Abstract: We derive sharp injectivity criteria for mappings $f:(-1,1)\rightarrow\mathbb{R} ^n$ in terms of Ahlfors' definition of the Schwarzian derivative for such mappings.

References:

[Ah]
L. V. Ahlfors, Cross-ratios and Schwarzian derivatives in $\mathbb{R} ^n$, Complex Analysis, 1-15, Birkhäuser, Basel, 1988. MR 90a:30055

[Ch]
M. Chuaqui, On a theorem of Nehari and quasidiscs, Ann. Acad. Sci. Fenn., Ser. A.I. Math. 18 (1993), 117-124. MR 94f:30026

[Ge-Po]
F. W. Gehring and Ch. Pommerenke, On the Nehari univalence criterion and quasicircles, Comment. Math. Helv. 59 (1984), 226-242. MR 85f:30023

[L]
D. Laugwitz, Differential and Riemannian Geometry, Academic Press, New York, 1965. MR 30:2406

[Ne 1]
Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. MR 10:696e

[Ne 2]
Z. Nehari, Univalence criteria depending on the Schwarzian derivative, Illinois J. Math. 23 (1979), 345-351. MR 80i:30033

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Additional Information:

Martin Chuaqui
Affiliation: Facultad de Matemáticas, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
Email: mchuaqui@mat.puc.cl

Julian Gevirtz
Affiliation: Facultad de Matemáticas, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
Email: jgevirtz@mat.puc.cl

DOI: 10.1090/S0002-9939-03-07013-8
PII: S 0002-9939(03)07013-8
Keywords: Schwarzian derivative, Nehari criteria, injectivity, extremal function
Received by editor(s): January 3, 2002
Received by editor(s) in revised form: September 5, 2002
Posted: June 12, 2003
Additional Notes: Both authors were partially supported by Fondecyt Grants # 1000627 and # 7000627
Communicated by: Juha M. Heinonen
Copyright of article: Copyright 2003, American Mathematical Society

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