Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

Antonia Jabbour; Stéphane Sabourau

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arXiv · 2009.10144

Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area

Antonia Jabbour·Stéphane Sabourau

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Abstract

We establish sharp universal upper bounds on the length of the shortest closed geodesic on a punctured sphere with three or four ends endowed with a complete Riemannian metric of finite area. These sharp curvature-free upper bounds are expressed in terms of the area of the punctured sphere. In both cases, we describe the extremal metrics, which are modeled on the Calabi-Croke sphere or the tetrahedral sphere. We also extend these optimal inequalities for reversible and non-necessarily reversible Finsler metrics. In this setting, we obtain optimal bounds for spheres with a larger number of punctures. Finally, we present a roughly asymptotically optimal upper bound on the length of the shortest closed geodesic for spheres/surfaces with a large number of punctures in terms of the area.

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BibTeX RIS

Antonia Jabbour, Stéphane Sabourau. 2020-09-21. Sharp upper bounds on the length of the shortest closed geodesic on complete punctured spheres of finite area. https://arxiv.org/abs/2009.10144

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