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.

中文翻译：

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有限区域的完整穿孔球体上最短闭合测地线长度的尖锐上限

我们在具有三个或四个末端的穿孔球体上的最短闭合测地线的长度上建立了明确的通用上限，该球体具有完整的有限面积黎曼度量。这些尖锐的无曲率上界以穿孔球体的面积表示。在这两种情况下，我们都描述了以 Calabi-Croke 球体或四面体球体为模型的极值度量。我们还将这些最优不等式扩展到可逆和非必要可逆的 Finsler 度量。在这种情况下，我们获得了具有大量穿刺的球体的最佳边界。最后，我们提出了在面积上具有大量穿孔的球体/表面的最短闭合测地线长度的大致渐近最优上界。