Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above

Abstract

Comparison and rigidity theorems are proved for curves of bounded geodesic curvature in singular spaes of curvature bounded above. Most of these estimates do not appear in the literature even for smooth curves in Riemannian manifolds. Geodesic curvature (which agrees with the usual one in the smooth case) is defined by comparison to curves of constant curvature in a model space. Two methods of comparison are used, preserving either sidelengths of inscribed triangles or arclength and chordlength. Using a majorization theorem of Reshetnyak, we obtain best possible global comparisons for arclength, chordlength, width and base angles in a CAT( K) space. A criterion for a metric ball to be a CAT( K) space is also given, in terms of the radius and the radial uniqueness property.