AN UPPER LENGTH ESTIMATE FOR CURVES IN CAT(K) SPACES
In Euclidean space, upper estimates for curvelength have been studied mostly in the previous century. Many of these have been extended over time, either to a larger class of spaces or to a larger class of curves. Due to limited tools, extentions to a larger class of spaces often end up with a restricted class of curves. With an appropriate variation of Reshetnyak’s fan construction technique in comparison geometry, the obstacle is overcome and a sharp upper length estimate for curves in terms of total curvature and the radius of a circumball are presented in this paper for CAT(K) spaces. The conﬁgurations of maximizers, which exist in standard spaces of constant curvature, are also completely determined. An interesting part is that in spaces of negative constant curvature, the maximizing conﬁgurations are totally diﬀerent from the case of nonnegative curvature.