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

Cheng, Herng Yi

doi:10.1090/tran/8653

Curvature-free linear length bounds on geodesics in closed Riemannian surfaces
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by Herng Yi Cheng PDF

Trans. Amer. Math. Soc. 375 (2022), 5217-5237

Abstract:

This paper proves that in any closed Riemannian surface $M$ with diameter $d$, the length of the $k$th-shortest geodesic between two given points $p$ and $q$ is at most $8kd$. This bound can be tightened further to $6kd$ if $p = q$. 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].

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