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The length of a shortest closed geodesic on a two-dimensional sphere and coverings by metric balls. (English) Zbl 1080.53031
Summary: In this paper we will present upper bounds for the length of a shortest closed geodesic on a manifold $$M$$ diffeomorphic to the standard two-dimensional sphere. The first result is that the length $$l(M)$$ of a shortest closed geodesic is bounded from above by $$4r$$, where $$r$$ is the radius of $$M$$. (In particular this means that $$l(M)$$ is bounded from above by $$2d$$, if $$M$$ can be covered by a ball of radius $$d/2$$, where $$d$$ is the diameter of $$M$$.) The second result is that $$l(M)$$ is bounded from above by $$2(\max\{r_1,r_2\} + r_1 + r_2)$$, if $$M$$ can be covered by two closed metric balls of radii $$r_{1}, r_{2}$$ respectively. For example, if $$r_{1} = r_{2} = d/2$$, then $$l(M) \leq 3d$$. The third result is that $$l(M) \leq 2 (\max\{r_1,r_2,r_3\} + r_1 + r_2 + r_3)$$, if $$M$$ can be covered by three closed metric balls of radii $$r_{1}, r_{2}, r_{3}$$. Finally, we present an estimate for $$l(M)$$ in terms of radii of $$k$$ metric balls covering $$M$$, where $$k \geq 3$$, if these balls have a special configuration.

##### MSC:
 53C22 Geodesics in global differential geometry
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##### References:
 [1] Bott, R., Lectures on Morse theory, old and new, Bull. Amer. Math. Soc., 7, 331-358, (1982) · Zbl 0505.58001 [2] Croke, C. B., Area and the length of the shortest closed geodesic, J. Differential Geom., 27, 1-21, (1988) · Zbl 0642.53045 [3] Maeda, M., The length of a closed geodesic on a compact surface, Kyushu J. Math, 48, 9-18, (1994) · Zbl 0818.53064 [4] Nabutovsky, A. and Rotman, R.: The length of a shortest closed geodesic on a two-dimensional sphere, IMRN No. 23, 1211-1222. · Zbl 1003.53030 [5] Sabourau, S.: Filling radius and short closed geodesics of the sphere, Preprint. · Zbl 1064.53020 [6] Calabi, E.; Cao, J., Simple closed geodesics on convex surfaces, J. Differential Geom., 36, 517-549, (1992) · Zbl 0768.53019 [7] Almgren, F., The homotopy groups of the integral cycle groups, Topology, 1, 257-299, (1962) · Zbl 0118.18503 [8] Pitts, J.: Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, Math Notes 27, Princeton Univ. Press, Princeton, NJ and Univ. Tokyo Press, Tokyo, 1981. · Zbl 0462.58003 [9] Nabutovsky, A. and Rotman, R.: Volume, diameter and the minimal mass of a stationary 1-cycle, preprint, arXiv:math.DG/$$0201269 v^2$$.Available at http://arXiv.org/abs/math. DG/0201269.
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