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Filling radius and short closed geodesics of the $$2$$-sphere. (English) Zbl 1064.53020
Let $$(M,g)$$ be a closed connected Riemannian manifold, and let $$d_g$$ be the distance on $$M$$. The map $$i: (M, d_g)\to (L^{\infty}(M), \|\cdot\|)$$ defined by $$i(x)(\cdot)=d_g(x, \cdot)$$ is an isometric embedding. Put $$U_{\delta}(M)$$ to be the $$\delta$$-tubular neighborhood of $$i(M)$$ in $$L^{\infty}(M)$$. The filling radius is $$\text{FillRad}(M)=\inf \{\delta >0| (i_{\delta})_*([M])=0\in H_n(U_{\delta}(M))\}$$.
The author shows that if $$M$$ is a Riemannian $$2$$-sphere and scg$$(M)$$ is the length of the shortest closed geodesic on $$M$$, then $$\text{FillRad}(M)\geq \frac{1}{12}\text{scg}(M)$$. He also shows that $$\text{FillRad}(M)\geq \frac{1}{20}\overline L(M)$$, where $$\overline L(M)$$ is the length of the shortest nontrivial curve among the simple closed geodesics of index zero or one and the figure-eight geodesics of null index. As a corollary the author gets that Area$$(M)\geq \frac{1}{20^2}\overline L(M)^2$$. He also proves that Area$$(M)\geq \frac{1}{C_1^2} \text{scg} (M)^2$$ and that $$\operatorname{Diam}(M)\geq \frac{1}{C_2} \text{scg}(M),$$ with $$C_1=12$$ and $$C_2=4$$.
This improves previous results of C. Croke who proved these inequalities with $$C_1=31$$ and $$C_2=9$$ [see “Area and the length of the shortest closed geodesic”, J. Differ. Geom. 27, No. 1, 1–21 (1988; Zbl 0642.53045)]. After having written the final version of the paper, the author learned that A. Nabutovsky and R. Rotman have independently proved the author’s inequality with $$C_2=4$$ and proved a sharper estimate with $$C_1=8$$ for the diameter of $$M$$, see their work [“The length of the shortest closed geodesic on a 2-dimensional sphere”, Int. Math. Res. Not. 2002, No. 23, 1211–1222 (2002; Zbl 1003.53030)].

##### MSC:
 53C20 Global Riemannian geometry, including pinching
##### Keywords:
filling radius; closed geodesics; $$1$$-cycles
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