AN UPPER LENGTH ESTIMATE FOR CURVES IN CAT(K) SPACES

Abstract

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