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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
     

Average curvature of convex curves in $H^2$

Author(s): Martin Bridgeman
Journal: Proc. Amer. Math. Soc. 126 (1998), 221-224.
MSC (1991): Primary 51M09, 52A55; Secondary 52A38, 52A15
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Abstract | References | Similar articles | Additional information

Abstract: A well-known result states that, if a curve $\alpha$ in $H^2$ has geodesic curvature less than or equal to one at every point, then $\alpha$ is embedded. The converse is obviously not true, but the embeddedness of a curve does give information about the curvature. We prove that, if $\alpha$ is a convex embedded curve in $H^2$, then the average curvature (curvature per unit length) of $\alpha$, denoted $K(\alpha)$, satisfies $K(\alpha) \leq 1$. This bound on the average curvature is tight as $K(\alpha)=1$ for $\alpha$ a horocycle.


References:

[B]
M. Bridgeman, Average bending of boundaries of convex cores, In preparation.
[S]
M. Spivak, A Comprehensive Introduction to Differential Geometry, Volume III, Publish or Perish (1979). MR 82g:53003c


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Additional Information:

DOI: 10.1090/S0002-9939-98-04047-7
PII: S 0002-9939(98)04047-7
Received by editor(s): June 13, 1996
Communicated by: Ronald A. Fintushel
Copyright of article: Copyright 1998, American Mathematical Society


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