Curvature-free linear length bounds on geodesics in closed Riemannian surfaces

Abstract

This paper proves that in any closed Riemannian surface M M with diameter d d , the length of the k k th-shortest geodesic between two given points p p and q q is at most 8 k d 8kd . This bound can be tightened further to 6 k d 6kd if p = q p = q . This improves prior estimates by A. Nabutovsky and R. Rotman [J. Differential Geom. 89 (2011), pp. 217–232; J. Topol. Anal. 5 (2013), pp. 409–438].