Average curvature of convex curves in $H^2$

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.