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Proceedings of the American Mathematical Society

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

 
 

 

Average curvature
of convex curves in $H^2$


Author: Martin Bridgeman
Journal: Proc. Amer. Math. Soc. 126 (1998), 221-224
MSC (1991): Primary 51M09, 52A55; Secondary 52A38, 52A15
DOI: https://doi.org/10.1090/S0002-9939-98-04047-7
MathSciNet review: 1415576
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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 [Enhancements On Off] (What's this?)

  • [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: https://doi.org/10.1090/S0002-9939-98-04047-7
Received by editor(s): June 13, 1996
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1998 American Mathematical Society

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