On average curvatures of convex curves in surfaces (Q1429187)

Considered are convex curves on a complete two-dimensional Riemannian manifold \(M\). The average curvature \(K( \alpha) \) of a convex curve \(\alpha\) in \(M\) is its geodesic curvature divided by its length. It is also defined if the length of \(\alpha\) is infinite. Let \(M\) be a simply connected Riemannian manifold with curvature bounded by numbers \(k_{1}\) and \(k_{2}\), where \(k_{1}\leq k_{2}\) and \(k_{1}\leq0\). For a convex curve \(\alpha\) on \(M\), let \(\delta( \alpha) \) denote the supremum of distances in \(M\) between arbitrary pairs of points on \(\alpha\). The authors give an estimate from above of the average curvature of \(\alpha\) in terms of the length of \(\alpha\), \(k_{1}\) and \(k_{2}\), provided that \(\delta\left( \alpha\right) \leq\pi/2\sqrt{k_{2}}\) when \(k_{2}>0\,\). The authors give several applications of their theorem for spaces of constant curvature.